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Journal of Mathematical Sciences, Vol. 256, No. 3, July, 2021
AROUND THE BAER–KAPLANSKY THEOREM
P. A. Krylov, A. A. Tuganbaev, and A. V. Tsarev UDC 512.541
Abstract. Using examples of modules and a number of familiar Ab elian groups, we demonstrate the
Kaplansky method of proving isomorphism theorems for endomorphism rings.
Keywords and phrases:Abelian group, endomorphism ring, isomorphism theorem for endomorphism
rings, Baer–Kaplansky theorem, Kaplansky method.
AMS Subject Classification:16D10, 16D70, 20K30
CONTENTS
Introduction............................................ 278
1.SomeDefinitionsandNotationinAbelianGroupTheory .................. 279
2.PrimaryPropertiesofEndomorphismsofAbelianGroups .................. 282
3.FiniteTopology.......................................... 284
4.TheCaseofVectorSpaces.................................... 285
5.Baer–KaplanskyTheorem.................................... 287
6.TopologicalIsomorphismsofEndomorphismRings...................... 289
7. Definability of p-groupsbyRadicalofEndomorphismRings................. 292
References............................................. 297
Introduction
One of the central problems concerning endomorphism rings is the question of how much the en-
domorphism ring defines an Abelian group or a module. In the simplest formulation, the results that
positively solve this problem (they are usually called the isomorphism theorems for endomorphism
rings) have the following form: if End A∼
=End B,thenA∼
=B. A stronger formulation of the iso-
morphism theorem has the following form: a given ring isomorphism ψ:EndA→End Bis induced
by some group or module isomorphism ϕ:A→B, i.e., ψ(h)=ϕhϕ−1,h∈End A.(Inthecaseof
modules, as a rule, semilinear module isomorphisms arise.) Theorems of this kind are related to the
following problem: For which modules are all the automorphisms of their endomorphism rings inner
automorphisms?
On endomorphism rings, we can define the finite topology and consider continuous isomorphisms
of endomorphism rings. They contain more information about the original modules. In this case, we
deal with topological isomorphism theorems.
This paper is a survey. Its goal is to give a detailed account of several characteristic isomorphism
theorems for endomorphism rings of modules and Abelian groups. What is noteworthy, to prove these
theorems, one of the modifications of the Kaplansky method is applied.
Sections 1 and 2 contain some necessary information about Abelian groups and their endomorphism
rings. Here we accept agreement on notation and terms.
In Sec. 3, we consider the finite topology on the endomorphism rings. In the remaining Secs. 4–
7, isomorphism theorems for endomorphism rings are considered. Moreover, in Sec. 4, we consider
the classical case of vector spaces over division rings and fields. Here there is the most transparent
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie
Obzory, Vol. 159,Algebra, 2019.
278 1072–3374/21/2563–0278 c
2021 Springer Science+Business Media, LLC
DOI 10.1007/s10958-021-05428-w
situation for this problem. In Sec. 5, we prove the Baer–Kaplansky theorem for p-groups. It can serve
as a standard for theorems of this kind. In Sec. 6, we prove some theorems of topological isomorphism.
In the last section, we show that for p-groups with unbounded basic subgroups, in the Baer–Kaplansky
theorem, one can confine ourselves to an isomorphism between Jacobson radicals of endomorphism
rings.
1. Some Definitions and Notation in Abelian Group Theory
The word “group” means an Abelian group with additive notation, with the exception of the
automorphism group. Most often, we denote the group by the letters Aor G.
In Abelian group theory, direct sums and direct products play a very important role. The direct
sum of the groups A1,A
2,...,A
nis denoted by A1A2...Anor
n
i=1
Ai.Wedenoteby
i∈I
Aithe
direct sum of the groups Ai,i∈I.
For an elem e nt aof the group A, the least positive integer nwith na = 0 is called the order of a;it
is denoted by o(a). If such an integer ndoes not exist, then we set o(a)=∞and say that the element
ais of infinite order.
Each Abelian group belongs to one of the following three classes of groups: periodic groups, torsion-
free groups, and mixed groups. In a periodic group, every element is of finite order; in a torsion-free
group, on the contrary, all nonzero elements are of infinite order. A mixed group contains both nonzero
elements of finite order and elements of infinite order. In a mixed group A, the subgroup t(A) consisting
of all elements of finite order is called the per iodic part or the peri odic subgroup of the group A.
AgroupAis called a primary group or a p-group if the order of any element of Ais a power of a
fixed prime integer p. A torsion group Ais equal to the direct sum of p-groups tp(A) for various p;the
groups tp(A) are called p-components of the group A.IfAis a mixed group, then the p-component of
its torsion part t(A) is called the p-component of A.
There are several known classes of groups which are direct sums of cyclic groups. A group is said to
be bounded if the orders of all its elements are bounded by some positive integer. A bounded group is
the direct sum of cyclic p-groups. If every element of a torsion group Ais of order not divisible by a
square of a positive integer, then Ais called an elementary group. An elementary group is the direct
sum of cyclic groups of prime orders.
For a group A, a subgroup Hof Ais said to be fully invariant if αH ⊆Hfor every endomorphism
αof the group A. The torsion part of the group and its p-components are fully invariant subgroups.
For every n∈N,weset
nA ={na |a∈A},A[n]={a∈A|na =0}.
The subgroups nA and A[n] are fully invariant in A.
Let pbe some prime integer, Abeagroup,andleta∈A. The largest nonnegative integer ksuch
that the equation pkx=ahas a solution in Ais called the p-height hA
p(a) of the element ain the
group A. If the equation pkx=ais solvable for any positive integer k,thenais called an element
of infinite p-height, hA
p(a)=∞.IfAand pare clear, then we write hp(a)andh(a)andcallh(a)the
height of the element a.
One says that a subgroup Bof the group Ais pure (in A) if the equation nx =b∈B, which has a
solution in the group A, also has a solution in the subgroup B. The subgroup Bis pure if and only if
B∩nA =nB for all n∈Z.
We give some properties of free or divisible groups. A free group is the direct sum of some number
of copies of the group Z.Adivisible group Dis the direct sum of quasi-cyclic groups Zp∞for various
pand copies of the group Q,
D=
p
mp
Zp∞
m0
Q.
279
The cardinal numbers m0and mpover all pform a complete independent system of invariants of the
divisible group D. A group is said to be reduced if it does not have a nonzero divisible subgroup. Every
group is representable in the form A=DV,whereDis a divisible group and Vis a reduced group.
The subgroup Dis uniquely determined; it is called the divisible part of the group A. The subgroup V
is called the reduced part of the group A. It is uniquely determined up to isomorphism. Every group
Acan be embedded in a divisible group Eas a subgroup, and Edoes not have a proper divisible
subgroup containing A. Such a group Eis called the divisible hull of the group A. Any two divisible
hulls E1and E2areisomorphicoverA, i.e., there exists an isomorphism E1→E2fixing the elements
of A.
A subgroup Bof the p-group Ais called a basis subgroup if the following three conditions hold:
(1) Bis the direct sum of cyclic groups;
(2) Bis a pure subgroup of the group A;
(3) the factor group A/B is a divisible group.
Every p-group contains basis subgroups. Any two basis subgroups are isomorphic.
Let Bbe a basis subgroup of the group A.Wehave
B=B1⊕B2⊕...⊕Bn⊕..., where Bn=
mn
Zpn.
For every n, there are direct decompositions
A=B1⊕B2⊕...⊕Bn⊕Anand An=Bn+1 An+1
In addition, the group Andoes not contain a cyclic direct summand of the order pn.
For p-groups, the following classical results of Kulikov are of great importance.
Theorem 1.1 (see [11, Theorem 27.5, Corollary 27.2]).Every bounded pure subgroup is a direct sum-
mand. In any p-group G, every element of order pand of finite height can be embedded in a cyclic
direct summand of G.
Therefore, if an element xof the p-group Ais of finite height, then there exists a decomposition
A=yYsuch that the element xhas a nonzero component in y.
In Abelian group theory, the Z-adic topology and the p-adic topology are often used. In the Z-adic
topology of the group A, the subgroups nA,n∈N, form a basis of neighborhoods of zero. In the
p-adic topology, the subgroups pkA,k0, form a basis. The structure of groups that are complete in
these topologies is known (see [11, Chap. 7], [18, Sec. 11]); such groups have a complete independent
system of invariants consisting of cardinal numbers. The class of groups that are complete in the Z-adic
topology coincides with the class of reduced algebraically compact groups.
For a torsion-free group A,therank r(A)ofAis the cardinality of some maximal linearly indepen-
dent system of elements of the group A,andthep-rank rp(A) of the group Ais the dimension of the
vector space A/pA over the field Zp,
rp(A)=dim
Zp(A/pA).
We have the inequality
rp(A)r(A).
For a mixe d g r o u p A, the rank of the factor group A/t(A) is called the torsion-free rank of the mixed
group A. Usually, one says simply the “rank” instead of the “torsion-free rank.” A torsion-free group
Gis of rank 1 if and only if Gis isomorphic to some subgroup of the group Q.
The tensor product of torsion-free groups also is a torsion-free group. For a torsion-free group A,
the tensor product AQis a Q-space and
r(A)=dim
Q(AQ).
280
We have a group embedding A→AQ,a→ a1, a∈A. The group AQis called the divisible
hull of the group A.
For torsion-free groups, there are very useful notions related to divisibility of elements by prime
integers. Let p1,p
2,...,p
n,... be a sequence of all prime integers which is ordered in ascending order.
The sequence of p-heights
χ(a)=hp1(a),h
p2(a),...,h
pn(a),...
is called the characteristic of the element aof the torsion-free group A.OnewritesχA(a)ifonewants
to specify the group in which the p-height and characteristics are calculated.
Any ordered sequence (k1,k
2,...,k
n,...) of nonnegative integers and the symbols ∞is called a
characteristic. Characteristics can be compared. Namely, we assume
k1,k
2,...,k
n,...l1,l
2,...,l
n,...,
if knlnfor all n∈N.Therelationturns the set of all characteristics into a complete lattice.
Now we introduce an equivalence relation on the set of all characteristics which leads to some
basic concept for torsion-free groups: the concept of type. Two characteristics (k1,k
2,...,k
n,...)and
(l1,l
2,...,l
n,...)aresaidtobeequivalent if kn=lnonly for a finite number of the subscripts nand,
for such n,thesymbolsknand lnare finite.
In the set of characteristics, equivalence classes are called types.Ifχ(a)∈tfor some type t, then one
says that the element ais of type tand we write t(a)=tor tA(a)=t.Ifχ(a)=(k1,k
2,...,k
n,...),
then we write
t(a)=(k1,k
2,...,k
n,...)=(ki),
i.e., any type is represented by the characteristic belonging to this type. The ordering on the set of
characteristics induces the ordering on the set of types.
A torsion-free group A,inwhichallnonzeroelementsthesametypet,issaidtobehomogeneous of
type t. In addition, we write t(A)=t. A group of rank 1 is homogeneous. By one of Baer’s theorems,
two torsion-free groups A,Bof rank 1 are isomorphic if and only if t(A)=t(B).
A torsion-free group Ais said to be fully decomposable if it is the direct sum of groups of rank 1.
Any two decompositions of the fully decomposable group into the direct sum of groups of rank 1 are
isomorphic. Let A=
i∈I
Aibe a fully decomposable torsion-free group, r(Ai)=1,i∈I. For every
type t,wedenotebyAtthe direct sum of all groups Aiof type t;ifAdoes not have such At,then
At= 0. The ranks r(At), where truns over the set of all types, form a complete independent system
of invariants of the group A.WesetΩ(A)={t(Ai)|i∈I}. The decomposition
A=
t∈Ω(A)
At
is called the canoni cal decomposition of the group A.
In comparison with completely decomposable groups, separable groups form a broader class. A
torsion-free group Ais said to be separable if every finite subset of elements of Ais contained in some
fully decomposable direct summand of the group A.
Abelian groups and modules over the ring of integers Zcannot be distinguished; they are the same
objects. The Abelian group theory can be considered a branch of module theory in which the specificity
of the ring Zis used. All concepts and constructions of the theory of modules are applicable to Abelian
groups.
In the theory of Abelian groups, various other rings also occur, in addition to the ring Z.Aglanceat
Abelian groups as modules over some rings is sometimes useful. Thus, divisible torsion-free groups are
precisely vector spaces over the field Q. The modules over the ring of residue classes Zpnare bounded
p-groups whose orders do not exceed pn.
From the point of view of the Abelian group theory, the most important ring is the ring of p-adic
integers
Zp.Itisknownthatp-groups and primary
Zp-modules coincide. The groups complete in the
281
p-adic topology are naturally
Zp-modules. The ring
Zpis an example of a complete discrete valuation
domain. Theories of abelian groups and modules over discrete valuation domains are related theories.
The latter is described in some detail in the book [18].
2. Primary Properties of Endomorphisms of Abelian Groups
If Aand Bare two Abelian groups, then Hom(A, B) is the group of all homomorphisms from A
into B,andEndA(sometimes End(A)) is the endomorphism ring of the group A.
We give some elementary properties of endomorphism rings. All of them are valid for endomorphism
rings of arbitrary modules.
Let A=BCbe some direct decomposition of the group A.Theprojection of the group Aonto
the direct summand Bwith kernel Cis the homomorphism π:A→B, which is defined as follows. If
a∈Aand a=b+c,whereb∈Band c∈C,thenπ(a)=b.Wedenotebyi:B→Athe embedding
of the group Bin the group A.Theniπ ∈End Aand (iπ)2=iπ, i.e., iπ is an idempotent of the
ring End A. It is called an idempotent endomorphism of the group A.Wesetε=iπ and identify
εwith π. Thus, we assume that the projection πis an endomorphism of Aacting on Bidentically
and annihilating C. It is clear that 1 −εalso is an idempotent orthogonal to ε. In addition, B=εA
and C=(1−ε)A=kerε, whence A=εA (1 −ε)A. The obtained decomposition holds for any
idempotent εof the ring End A.
More generally, if A=A1A2... Anis some direct decomposition of the group A,thenwe
denote the projection A→Aiwith kernel
j=i
Ajby εiand obtain Ai=εiA(i=1,2,...,n); in
addition, {εi|i=1,2,...,n}is a complete orthogonal system of idempotent endomorphisms of the
group A.
Proposition 2.1. There is a bijective correspondence
A=ε1A...εnA→ End A=(EndA)ε1...(End A)εn
between finite direct decompositi ons of the group Aand decompositions of the ring End Ainto direct
sums of left ideals, where {εi|i=1,2,...,n}is a complete orthogonal system of idempotents of the
ring End A.
Proof. We have already proved that for a given direct decomposition A=A1A2...An,thereexists
acompletesystem{εi|i=1,2,...,n}of orthogonal idempotents of the ring End Asuch that Ai=εiA
for all i. This system leads to the decomposition End A=(EndA)ε1(End A)ε2...(End A)εnof
the ring End Ainto the direct sum of left ideals.
Conversely, if End A=L1L2...Ln,whereLiare left ideals of the ring End A,thenwewrite1=
ε1+ε2+...+εn,εi∈Li, and obtain a complete orthogonal system {εi|i=1,2,...,n}of idempotents
of the ring End A. It is easy to verify that we have the decomposition A=ε1Aε2A...εnA.The
constructed correspondence is bijective.
We consider several standard relations between a group and its endomorphism ring related to
idempotent endomorphisms. The following properties directly follow from Proposition 2.1.
(a) If εis an idempotent of the ring End A,thenεA is an indecomposable direct summand of the group
Aif and only if εis a primitive idempotent.
(b) Let εand ωbe two idempotents of the ring End A. There exist canonical group isomorphisms
Hom(ωA, εA)∼
=ε(End A)ωand canonical ring isomorphisms End(εA)∼
=ε(End A)ε.
Indeed, let ϕ:ωA →εA be some homomorphism. It can be extended to an endomorphism ϕof the
group Asuch that ϕannihilates the complement summand (1 −ω)Aof ωA. We obtain the required
isomorphism f:Hom(ωA, εA)→ε(End A)ωfrom the correspondence ϕf
−→ εϕω. Indeed, if εψω ∈
ε(End A)ωfor some ψ∈End A,thenεψω|ωA is a homomorphism ωA →εA and f:εψωωA → εψω.
For ε=ω, we have the isomorphism End(εA)∼
=ε(End A)ε; this is a ring isomorphism.
282
Let A=BCand ε:A→Bbe the projection with kernel C. We can assume that End B
is a subring of the ring End Aif we identify End Bwith ε(End A)εby the use of the isomorphism
constructed in the property (b).
We consider two primary facts on relations between isomorphisms of groups and isomorphisms of
their endomorphism rings.
(c) If two groups Aand Care isomorphic, then their endomorphism rings are isomorphic. More
precisely, every group isomorphism ϕ:A→Cinduces the ring isomorphism ψ:EndA→End C
which acts by the rule ψ:η→ ϕηϕ−1,η ∈End A.
For η1,η
2∈End A,wehavetherelations
ψ(η1+η2)=ψ(η1)+ψ(η2),ψ(η1η2)=ψ(η1)ψ(η2),ψ(idA)=idC.
Consequently, ψis a ring homomorphism. Further, if 0 =η∈End A, then it is clear that ϕηϕ−1=0,
i.e., ker ψ=0.Nowletξ∈End C;thenψ(ϕ−1ξϕ)=ξ; therefore, ψis a ring isomorphism.
(d) Let A=A1A2and Cbe groups. If ψ:EndA→End Cis a ring isomorphism, then the group
Chas the decomposition C=C1C2,whereψinduces isomorphisms End Ai→End Ci,i=1,2.
We denote by εthe projection A→A1with kernel A2.Thenω=ψ(ε) is an idempotent of the
ring End C.WehavetherelationC=C1C2,whereC1=ωC and C2=kerω. The isomorphism ψ
induces the ring isomorphism ε(End A)ε→ω(End C)ωand, therefore, the ring isomorphism End A1→
End C1; see property (b). The second isomorphism End A2→End C2can be proved similarly.
In some cases, the endomorphism ring can be easily calculated. For example, we have End Z∼
=Z,
End Zn∼
=Zn,EndQp∼
=Qp,whereQp={s/t ∈Q|(t, p)=1},EndQ∼
=Q,EndZp∞∼
=End
Zp∼
=
Zp.
By considering direct sums of groups, we can obtain examples of endomorphism rings in the matrix
form. First, we consider the corresponding construction.
Let us have the direct sum of groups A=
n
i=1
Ai. We construct the square matrix
αji=⎛
⎜
⎜
⎝
α11 α12 ... α
1n
α21 α22 ... α
2n
... ... ... ...
αn1αn2... α
nn
⎞
⎟
⎟
⎠
with elements αji ∈Hom(Ai,A
j). For such matrices, one can define ordinary matrix operations of
addition and multiplication. It is easy to see that addition and multiplication of matrices are always
feasible and lead to matrices of the same form. As a result, we obtain a ring of matrices of the indicated
type; such rings are called formal matrix rings or generalized matrix rings, (see [19–21]). The standard
isomorphism from the ring of operators of a finite-dimensional vector space onto the matrix ring in
the case of Abelian groups (and modules) takes the following form.
Proposition 2.2. The endomorphism ring of the group A=
n
i=1
Aiis isomorphic to the ring of all
matrices αjiof order n,whereαji ∈Hom(Ai,A
j).
Now we can continue the list of endomorphism rings. Namely, there are isomorphisms
End(ZQ)∼
=Z0
QQ
,End(ZnZ)∼
=ZnZn
0Z,
End(ZpnZpm)∼
=ZpnZpn
ZpnZpm(n<m),End(Zp∞Q)∼
=
Zp
Qp
0Q,
where
Qpis the field of p-adic numbers.
283
The part of algebra that studies endomorphism rings of Abelian groups can also be referred to
the theory of Abelian groups and to the theory of endomorphism rings of arbitrary modules. The
monograph [17] is entirely devoted to various aspects of the theory of endomorphism rings of Abelian
groups.
3. Finite Topology
On the ring of endomorphisms of any module, we can define one very useful topology. This is the
so-called finite topology, which is an example of a linear topology. A linear topology on a module (ring)
is a topology for which there is a basis of neighborhoods of zero consisting of submodules (left ideals)
and the corresponding residue classes form a basis of open sets. Examples of linear topologies are the
Z-adic topology and the p-adic topology on Abelian groups.
Let Mbe a right module over by some ring and End Mits endomorphism ring. The finite topology
is defined on the ring End Mwith the use of the following subbasis of neighborhoods of zero:
Ux={α∈End M|α(x)=0},
where xruns over all the elements of the module M. It is clear that Uxare left ideals of the ring
End M. The ideals
UX={α∈End M|αX =0}
form a basis of neighborhoods of zero where Xruns over all finite subsets of the module M.Sincewe
have UX=
x∈X
Ux, the residue classes α+UX, for all finite subsets Xof the module M,formabasis
of neighborhoods of the element α∈End M. The finite topology is always a Hausdorff topology. We
formulate the main theorem about the finite topology.
Theorem 3.1. The endomorphism ring of any module Mis a topological ring which is complete in
the finite topology.
Proof. Since Uxare left ideals, it is obvious that the addition and the subtraction are continuous in the
ring End M. Now we verify that the multiplication is continuous. We take arbitrary endomorphisms
α, β ∈End M,andletαβ +Uxbe a neighborhood of the element αβ.SinceUβ(x)β⊆Ux,wehave
(α+Uβ(x))(β+Ux)⊆αβ +Ux,
which implies that the multiplication is continuous.
Thus, End Mis a topological ring. We prove this ring is complete. We assume that {αi}i∈Iis a
Cauchy sequence in ring End M. By the definitions of the finite topology and a Cauchy sequence,
the set of subscripts Iis ordered with respect to the order which is dual to the order on the finite
subsets of the module M. For a given element x∈M, there exists a subscript i0∈Isuch that
αi−αj∈Uxfor all i, j > i0. This means that αi(x)=αj(x) for quite large subscripts iand j.There
exists an endomorphism αof the module Msuch that α(x) the common value of all such αi(x). Then
α−αi∈Uxfor i>i
0.Thus,αis the limit of this Cauchy sequence {αi}i∈I. We obtain that every
Cauchy sequence converges in the ring End M; this means that End Mis complete.
When applying the finite topology, the most important are the completeness of the ring endomor-
phisms and continuous isomorphisms between endomorphism rings.
Sometimes, the finite topology can be defined in terms of the endomorphism ring itself.
Proposition 3.2. The following assertions hold.
1. Let Vbe a vector space over some division ring. The finite topology of the operator ring End V
of the space Vwill be defined if we take the set of left annihilators of primitive idempotents as a
subbasis of neighborhoods of zero.
284
2. Let Gbe a reduced p-group. The finite topology of the ring End Gcan be defined if we take the set
of left annihilators of elements αε as a subbasis neighborhoods of zero, where α∈End Gand εis
a primitive idempotent. If the group Gdoes not have elements of infinite height, then it suffices to
take only left annihilators of primitive idempotents.
3. If Gis a separable torsion-free group, then the finite topology of the ring End Gcan be similarly
defined by taking the set of left annihilators of primitive idempotents as a subbasis of neighborhoods
of zero.
Proof. 1. Let xbe an arbitrary vector of the space V,Xbe the subspace in Vgenerated by the vector
x,andletε:V→Xbe the projection. The relation Ux=(EndV)(1 −ε)meansthatUxis a left
annihilator of the primitive idempotent ε.
2. For every element x∈G, there exists of cyclic direct summand yof the group Gwith o(x)
o(y). Consequently, there exists an endomorphism α∈End Gwhich maps yonto x.Letε:G→y
be the projection. Then Uxcoincides with the left annihilator of the element αε and εis a primitive
idempotent. If the group Gdoes not have elements of infinite height, then for the finite topology, the set
of left ideals Uxis a subbasis of neighborhoods of zero, where element xruns over only elements such
that xis a direct summand of the group G.Ifε:G→xis the projection, then Ux=(EndV)(1−ε),
similarly to item 1.
3. Similarly to item 2, a subbasis of neighborhoods of zero can be defined as the set of left ideals
Ux,wherexruns over only elements such that x∗is a direct summand of the group G,wherex∗
is a pure subgroup generated by the element x.
4. TheCaseofVectorSpaces
Of course, isomorphic groups have isomorphic endomorphism rings. In general, the converse problem
is much more difficult. In its most general form, it can be formulated as follows: how are two groups
connected if their endomorphism rings are isomorphic to each other? For example, will these groups be
isomorphic? The natural formulation of this problem is suggested by the property (c) in Sec. 2. Will a
given ring isomorphism ψ:EndA→End Bbe induced by some group isomorphism ϕ:A→B, i.e.,
whether the formula ψ(η)=ϕηϕ−1holds for all η∈End A?
When passing to modules, new versions of the formulation of the problem arise. Thus, we can
consider modules over different rings. In this situation, semilinear isomorphisms of modules are used.
Let Rand Sbe two rings, Abe a right R-module, and Bbe a right S-module. An additive
isomorphism ϕ:A→Bis called a semilinear isomorphism of the modules Aand Bif there is a ring
isomorphism τ:R→Ssuch that ϕ(ar)=ϕ(a)τ(r) for all a∈Aand r∈R. In addition, one says
that a ring isomorphism (or an algebra isomorphism) of endomorphism rings ψ:End
RA→EndSB
is induced by a semilinear isomorphism ϕ:A→Bif ψ(η)=ϕηϕ−1,η∈EndRA.
This section and the following two sections are directly related to the definability problem of a
group (module) by its endomorphism ring. First of all, we turn to the case of vector spaces and their
operator rings. By virtue of the simple (from the point of view of module theory) structure of vector
spaces, a number of ideas and methods, which we will continue to apply here, appear in a rather simple
and direct form.
We will consider the right vector spaces over division rings.
Theorem 4.1 (Baer [2]).Let Vand Wbe two vector spaces over division rings Dand F, respec-
tively. Then every ring isomorphism from the operator ring EndDVonto EndFWis induced by some
semilinear isomorphism from Vonto W.
Proof. Let ψ:End
DV→EndFWbe a ring isomorphism. Further, for convenience, we write α∗
instead of ψ(α), where α∈EndDV. We fix a nonzero vector a∈V.Letε:V→aD be the projection,
where aD is the one-dimensional subspace generated by the vector a.ThenaD =εV .Sinceεis an
285
idempotent of the ring EndDV,wehavethatε∗is an idempotent of the ring EndFW.Consequently,
ε∗Wis an one-dimensional direct summand of the space W.Letbbe some nonzero vector in ε∗W.
Then bF =ε∗W. By the property (b) from Sec. 2, we identify the rings EndD(aD)andε(EndDV)ε;
we also identify the rings EndF(bF )andε∗(EndFW)ε∗.Consequently,ψinduces the ring isomorphism
EndD(aD)→EndF(bF ).
Note the following fact. For a fixed element d∈D, there exists a unique operator σd∈EndD(aD)
which acts by the rule σd(ad)=add,whered∈D.Inparticular,σd(a)=ad. The correspondence d→
σddefines a ring isomorphism D→EndD(aD). There exists a similar isomorphism F→EndF(bF ).
We denote by τthe composition of isomorphisms
D→EndD(aD)ψ
−→ EndF(bF )→F
In addition, the relation σ∗
d(b)=bτ(d)holds.
We construct the required semilinear isomorphism from Vonto W. For an arbitrary vector x∈V
we take an operator αof the space Vsuch that x=α(a). We define the mapping ϕ:V→Wby
setting ϕ(x)=α∗(b). The mapping ϕis well defined, i.e., it is independent of the choice of operator
α. Indeed, if x=α1(a), α1∈EndD(V), then (α−α1)(a)=0and(α−α1)ε= 0. Therefore,
(α−α1)ε∗=(α∗−α∗
1)ε∗=0;
consequently, (α∗−α∗
1)(b)=0andα∗(b)=α∗
1(b).
We take another vector y∈Vand take β∈EndDVwith y=β(a). Then x+y=(α+β)(a);
therefore,
ϕ(x+y)=(α+β)∗(b)=α∗(b)+β∗(b)=ϕ(x)+ϕ(y).
Thus, ϕis an additive homomorphism.
We show that ϕis a semilinear mapping, i.e., we verify that the relation ϕ(xd)=ϕ(x)τ(d)holds
for any x∈Vand d∈D.Letx=α(a), as above. Then xd =α(a)d=α(ad)Sinceϕ(x)=α∗(b), we
have
ϕ(x)τ(d)=α∗(b)τ(d)=α∗bτ(d).
On the other hand, since xd =α(ad)=ασd(a), we have
ϕ(xd)=(ασd)∗(b)=α∗(σ∗
d(b)) = α∗bτ(d).
Thus, ϕ(xd)=ϕ(x)τ(d).
If ϕ(x)=α∗(b)=0forsomex∈V,then(αε)∗=α∗ε∗=0.Thisimpliesthatαε =0and
x=αε(a) = 0, i.e., ker ϕ= 0. For every vector z∈W, there exists an operator γ∈EndFWwith
z=γ(b). Let γ=α∗for some α∈EndDV.Thenz=γ(b)=α∗(b)=ϕ(x), where x=α(a). We
obtain that ϕis a bijection; in other words, ϕis a semilinear isomorphism.
We take an arbitrary operator μ∈EndDVand a vector z∈W. Then we take a vector x∈Vwith
z=ϕ(x) and an operator α∈EndDVwith x=α(a). Then z=ϕ(x)=α∗(b). Now we have the
relations
μ∗(z)=(μα)∗(b)=ϕμα(a)=ϕμ(x)=(ϕμϕ−1)(z).
Thus, ψ(μ)=μ∗=ϕμϕ−1, i.e., the isomorphism ψis induced by the isomorphism ϕ.
If the space Vis of finite dimension m, then the ring EndDVis isomorphic to the matrix ring
Mm(D)ofordermover the division ring D. Then we have the following partial case of Theorem 4.1.
Corollary 4.2. If the rings Mm(D)and Mn(F)are isomorphic, then m=nand the division rings
Dand Fare isomorphic.
We consider the situation where Dand Fare fields. We identify any element d∈Dwith its action
on V.Inshort,weassumethatDis the center of the ring EndDV. Similarly, we identify the field
Fwith the center of the ring EndFW. Under ring isomorphisms, the center passes onto the center.
Therefore, under the conditions of Theorem 4.1, we can assume that Vand Ware spaces over the same
286
field, say F. However, in this case we can only prove that the ring isomorphism ψ:End
FV→EndFW
is induced by some semilinear F-space isomorphism V→W.
Since the rings EndFVand EndFWare F-algebras, it is natural to go further and assume that ψis
an F-algebra isomorphism, i.e., ψ(sα)=sψ(α) for all s∈Fand α∈EndFV. Under this assumption,
taking into account the equalities ψ(s·idV)=sψ(idV)=s·idW, we obtain that the isomorphism
of F-algebras EndFV,End
FWacts identically on F. In general, the isomorphisms between the
endomorphism algebras EndFVand EndFWare precisely ring isomorphisms that leave the elements
of the center in place. Taking this into account, we can write down the following result.
Corollary 4.3. Let Vand Wbe two vector spaces over the fiel d F. Then every isomorphism of
endomorphism algebras EndFVand EndFWis induced by some isomorphism of spaces Vand W.
Isomorphism theorems have one important application. Let ψbe an automorphism of the F-algebra
EndFV. By Corollary 4.3, there exists an automorphism (invertible operator) ϕof the space Vsuch
that ψ(α)=ϕαϕ−1for every α∈EndFV. Since automorphisms of the space Vcoincide with
invertible elements of the algebra EndFV, the last relation means that ψis an inner automorphism
of the algebra EndFV.
Corollary 4.4. The following assertions hold.
1. Every automorphism of the algebra endomorphisms of the vector space over the field is inner.
2. (Skolem and Noether). Every automorphism of the matrix algebra Mn(F)is an inner automorphism.
We might say that Baer’s proof of Theorem 4.1 in [2] is of a geometric character. In the proof
of Theorem 4.1, we used the method which is called the Kaplansky method. Relevant reasoning first
appeared in his book [15]. The essence of Kaplansky’s method is as follows. The primitive idempotents
of the operator ring correspond to the indecomposable subspaces; they are one-dimensional in this case.
In order to construct an isomorphism from the space Vonto the space W, we transfer the properties
of such summands by means of operators in order to obtain necessary elements of the space W.In
one or another form, the Kaplansky method will be applied several times in the remaining sections of
this paper.
5. Baer–Kaplansky Theorem
We state and prove, perhaps, the most famous result on the definability of Abelian groups or
modules by their endomorphism rings. We are talking about the following remarkable theorem.
Theorem 5.1 (Baer [1], Kaplansky [15]).If Aand Care torsion groups with isomorphic endomor-
phism rings, then every ring isomorphism End A→End Cis induced by some group isomorphism
A→C.
Proof. We can restrict ourself to the case of p-groups. Indeed, we have
A=
p∈P
tp(A),C=
p∈P
tp(C),
where tp(A)andtp(C)arethep-components of the groups Aand C, respectively. Then
End A=
p∈P
End tp(A),End C=
p∈P
End tp(C).
Since
End tp(A)=
(n, p)=1
nEnd A, End tp(C)=
(n, p)=1
nEnd C,
every ring isomorphism End A→End Cmust map End tp(A)ontoEndtp(C). Therefore, it suffices to
assume that Aand Care p-groups.
287
Then we proceed with a fixed ring isomorphism ψ:EndA→End C. For every η∈End A,wewrite
ψ(η)=η∗.
If Ais a cyclic or quasi-cyclic p-group, then it is easy to see that A∼
=C(see Section 2 about the
structure of endomorphism rings of such groups). Then we divide the proof into three cases.
Case 1: Ais a bounded group. Then Ais the direct sum of cyclic p-groups. Let gbe one of the
generators of some cyclic direct summand of the group Aof the largest order pk.Ifε:A→gis
the projection, then εis an idempotent of the ring End Aand ε∗is an idempotent of the ring End C.
Consequently, ε∗Cis a direct summand of the group C. By the property (d) from Sec. 2, the iso-
morphism ψinduces the ring isomorphism Endg→End(ε∗C). Therefore, ε∗Cis a cyclic group h
of order pk. Now we can construct the required isomorphism ϕ:A→C. For any element a∈A,
we take an endomorphism η∈End Awith a=η(g) and define a mapping ϕ:A→Csuch that
ϕ(a)=η∗(h). Similarly to the proof of Theorem 4.1, we can verify that the mapping ϕis well defined,
ϕis an isomorphism and ϕinduces ψ.
Case 2: A=BD,whereBis a bounded group and Dis a nonzero divisible group. Let gbe a
cyclic direct summand of maximal order pkin the group B,Ebe a direct summand of the group D
which is isomorphic to the group Zp∞,andlet
E=d1,d
2,...,d
n,...,pd
1=0,pd
n+1 =dnfor n1.
We denote by ε:A→gand π:A→Ethe corresponding projections. Similarly to Case 1, we
obtain that ε∗Cis a cyclic direct summand of the group Cand π∗Cis a direct summand of the group
Cwhich is isomorphic to the group Zp∞. We define two groups ε∗Cand π∗Cwith the use of their
generators:
ε∗C=h,π
∗C=e1,e
2,...,e
n,...,pe
1=0,pe
n+1 =enfor n1.
We represent an arbitrary element a∈Ain the form a=a1+a2,wherea1∈B,a2∈D,andtake
an endomorphism η∈End Asuch that η(g)=a1,η(dn)=a2for some n. We construct a mapping
ϕ:A→Cby setting ϕ(a)=η∗(h+en). First, we show that ϕdoes not depend on the choice of η
and n.Wetakeη1∈End Asuch that η1(g)=a1and η1(dm)=a2, and we can assume that mn.
Then we obtain the relations
(η−η1)(g)=0,(pm−nη−η1)(dm)=0.
Therefore, (η−η1)ε= 0 and the endomorphism (pm−nη−η1)πannihilates E[pm]. Therefore, the
endomorphism (pm−nη−η1)πis divided by pm. Then the endomorphism (pm−nη−η1)π∗is also
divided by pm; consequently, it annihilates the element em. Thus, we obtain that η∗(h)=η∗
1(h)and
η∗(en)=pm−nη∗(em)=η∗
1(em), whence η∗(h+en)=η∗
1(h+em).
Similarly to Case 1, we refer to the proof of Theorem 4.1 for the verification of the property that ϕ
is an isomorphism inducing the isomorphism ψ.
Case 3: Ahas an unbounded basis subgroup. It follows from properties of basis subgroups (see Sec. 1)
that there exist decompositions
A=a1a2...akAk,k∈N,
such that Ak=ak+1Ak+1 and o(ak)=pnk,where1n1<n
2<... <n
k<....Letεk:A→ak
be the projection. For distinct subscripts jand k, we define an endomorphism γjk of the group Aas
follows. The endomorphism γjk maps the direct summand, which is complementary to akin the
above decomposition of A,onto0.Italsomapsfromakonto aj(respectively, onto pnj−nkaj)ifj<k
(respectively, j>k). Then
(1) γjkεk=γjk =εjγjk for all j=k;
(2) γkjγjk =p|nj−nk|εk, for all j=k;
(3) γijγjk =γik if i<j<k or i>j>k.
288
The endomorphisms ε∗
kand γ∗
jk of the group Calso satisfy conditions (1)–(3). The subgroups ε∗
kC
are cyclic direct summands of the group Cof orders coinciding with the orders of the groups εkA,by
the property (d) from Sec. 2. It follows from the condition (2) that the endomorphism γ∗
k,k+1 maps
ε∗
k+1Conto ε∗
kC.Wesetε∗
kC=ckand show that we can choose the generators ckto satisfy the
relations γ∗
k,k+1(ck+1)=ckfor all k. Indeed, if the elements c1,c
2,...,c
kare already chosen and the
element c
k+1 generates the subgroup ε∗
k+1C,thenγ∗
k,k+1(c
k+1)=tckfor some t∈Z. Further, it follows
from (2) that γ∗
k+1,k(tck)=pnk+1 −nkc
k+1; by considering the orders of the elements, that (p, t)=1.
We take the element ck+1 =sc
k+1,wherest ≡1(modpnk). Then γ∗
k,k+1(ck+1)=ck.Furthermore,it
follows from (3) that γ∗
jk(ck)=cjfor all j<k.
For an arbitrary element a∈A, we take an endomorphism η∈End Asuch that η(ak)=afor some
k∈N. We define the mapping ϕ:A→Cby setting ϕ(a)=η∗(ck). We verify that the mapping
ϕis well defined. Let η1(aj)=a,whereη1∈End Aand jk.Then(ηγkj −η1)εj= 0, whence
(η∗γ∗
kj −η∗
1)ε∗
j= 0; therefore, η∗(ck)=η∗
1(cj).
Finally, similarly to the proof of Theorem 4.1, we can verify that the constructed mapping ϕis an
isomorphism which induces the isomorphism ψ.
Similarly to the case of vector spaces, we obtain a corollary related to automorphisms of torsion
groups.
Corollary 5.2. For a torsion gro u p G, every automorphism of the endomorphism ring of Gis inner.
6. Topological Isomorphisms of Endomorphism Rings
On the endomorphism ring, we have the finite topology, so it is natural to consider isomorphisms of
endomorphism rings that are continuous in both directions. They more accurately determine the struc-
ture of the source module. We call such isomorphisms topological and discuss theorems of topological
isomorphism.
Thus, by a topological isomorphism ψ:EndA→End C, we mean a ring isomorphism ψsuch that
ψand ψ−1are continuous with respect to the finite topology. It can be directly verified that every
group isomorphism A→Cinduces a topological ring isomorphism End A→End C.Wealsopay
attention to the fact that the ring isomorphisms appearing in Theorems 4.1 and 5.1 are topological.
This follows from Proposition 3.2.
In this section, all groups are torsion-free. Most of the concepts related to torsion-free groups are
defined in Sec. 1. In addition, we recall that a type is said to be idempotent if it contains a characteristic
consisting of the symbols 0 and ∞. The type of torsion-free group Aof rank 1 is idempotent if and
only if Ais isomorphic to the additive group of some subring of the rational number field Q.
Every Abelian group Ais a natural left module over its endomorphism ring. Let Abe torsion free. In
this case, the Q-algebra End AQis called the quasi-endomorphism ring or the quasi-endomorphism
algebra of the group A. The action of the ring End Aon the group Aextends to the action of the
ring End AQon the divisible hull AQof the group A. Thus, we get the left (End AQ)-module
AQ.
We assume that the group Ais embedded in the Q-space AQby identifying an element a∈A
with the element a1. We accept the same agreement with respect to End Aand End AQ.
Pure fully invariant subgroups of the group Aare briefly called pf i-subgroups.Itiseasytoverify
that the correspondences
H→ HQ,W
→ W∩A
are mutually inverse isomorphisms between the lattice of pf i-subgroups of the group Aand the
submodule lattice of the (End AQ)-module AQ.
We recall that a group Ais said to be irreducible if it does not have a proper pf i-subgroup. The
irreducibility of the group Ais equivalent to the irreducibility of the (End AQ)-module AQ.
289
Definition 6.1. A torsion-free group Gis said to be fully transitive if for any its elements a, b =0
with χ(a)χ(b), there exists an endomorphism α∈End Gwith α(a)=b.
Homogeneous separable groups and algebraically compact groups are simplest examples of fully
transitive groups.
Lemma 6.2. A homogeneous fully transitive torsion-free group Gis irreducible. In addition, if Gis
of an idempotent type, then every its pure subgroup contains a generator of the End G-module G.
Proof. We assume that His a nonzero pfi-subgroup of the group G.Leta∈Hand b∈Gbe
some nonzero elements. We take a positive integer nwith χ(a)χ(nb). Then α(a)=nb for some
α∈End G. Since the subgroup His fully invariant, we have nb ∈H. Since the subgroup His pure,
b∈H.Consequently,H=G; therefore, Gdoes not contain a proper pfi-subgroup.
If the group Gis of idempotent type, then every its nonzero pure subgroup contains element a=0
with characteristic χ(a) consisting of 0 and ∞.Thenχ(a)χ(b) for any nonzero element b∈G.
Therefore, b∈(End G)aand (End G)a=G, i.e., the element agenerates the End G-module G.
Theorem 6.3 (see [16]).Let Gand Hbe two homogeneous fully transitive torsion-free groups whose
types are idempotent. Then every topological ring isomorphism between End Gand End His induced
by some group isomorphism between Gand H.
Proof. Let ψ:EndG→End Hbe some topological ring isomorphism. For convenience, we use the
following notation:
V=GQ,W=HQ,R=EndGQ,S=EndHQ.
Then Vis a faithful irreducible R-module and Wis a faithful irreducible S-module by Lemma 6.2.
Further, we set
D=End
RV, F =End
SW, K =End
DV, L =End
FW.
Here Dand Fare division rings by the Schur lemma. By the familiar density theorem of Jacobson–
Chevalley for irreducible modules, the ring Ris dense in the finite topology of the ring K,andthe
ring Sis dense in the finite topology of the ring L.
We identify the ring End G(respectively, End H) with its image under the canonical embedding
End G→R(respectively, End H→S). Then finite topology of the ring End G(respectively, End H)
coincides with the topology induced by the finite topology of the ring K(respectively, L). Therefore,
ψidQis a topological ring isomorphism of Rand Swhich is also denoted by ψ.SinceR(respec-
tively, S) is dense in the complete ring K(respectively, L), we have that ψcan be uniquely extended
to the ring isomorphism K→Lwhich is also denoted by ψ.Asabove,wewriteη∗instead of ψ(η).
Let gbe some fixed generator of the EndG-module Gwhich exists by Lemma 6.2. We denote by
Athe subspace of the D-space Vgenerated by the element gand consider the projection π:V→A.
Then π∈Kand π2=π. Hence (π∗)2=π∗and π∗:W→π∗Wis the projection. In addition,
D∼
=EndDA∼
=πKπ ∼
=π∗Lπ∗∼
=EndF(π∗W)
Consequently, dimF(π∗W) = 1; see the property (b) from Sec. 2 about the isomorphism EndDA∼
=
πKπ and a similar isomorphism for the ring L.Inπ∗W∩H,wetakesomeelementhgenerating the
End H-module H.
We define the mapping ϕ:G→Has follows. For an arbitrary element a∈G,wetakean
endomorphism η∈End Gwith a=η(g). We set ϕ(a)=η∗(h). We verify that the action of ϕdoes
not depend on the choice of endomorphism η.Ifa=η1(g), where η1∈End G,then(η−η1)(g)=0.
Consequently, (η−η1)π=0,sincetheD-space Ais one-dimensional. Hence (η∗−η∗
1)π∗=0and
(η∗−η∗
1)(h) = 0, i.e., η∗(h)=η∗
1(h). Thus, the mapping ϕis well defined.
It remains to verify that ϕis an isomorphism which induces the isomorphism ψ. In general, it
repeats the corresponding places in the proof of Theorem 4.1.
290
Of course, in an indirect form, the proof of Theorem 6.3 is based on the Kaplansky method.
If we apply Theorem 6.3 to a homogeneous fully transitive group, then we obtain the result which
is similar to Corollaries 4.4 and 5.2.
Corollary 6.4. For a homogeneous fully transitive torsion-free group of idempotent type, every topo-
logical automorphism of its endomorphism ring is inner.
We apply Theorem 6.3 to homogeneous separable torsion-free groups defined in Sec. 1. It is easy to
prove that such a group is fully transitive. In addition, it follows from Proposition 3.2(3) that every
isomorphism between endomorphism rings of two separable torsion-free groups is topological. Then
we have the following result.
Corollary 6.5. If Gand Hare two homogeneous separable torsion-free groups of idempotent types,
then every isomorphism End G→End His induced by some isomorphism G→H.
We also have the following assertion.
Corollary 6.6. Let Gand Hbe two fully decomposable torsion-free groups such that types of all
homogeneous components of these groups are idempotent. Then every isomorphism End G→End H
is induced by some isomorphism G→H.
Proof. Let the canonical decompositions of the groups Gand Hbe of the form
G=
t∈Ω(G)
Gt,H=
t∈Ω(H)
Ht,
where Gtand Htare so-called homogeneous components of the groups Gand H, respectively. Let
ψ:EndG→End Hbe some ring isomorphism. For every t∈Ω(G), we denote by εtthe projection
G→Gt. There are isomorphisms
End Gt∼
=εt(End G)εt∼
=ψ(εt)(End H)ψ(εt)∼
=Endψ(εt)H.
Since Gtis a homogeneous group of idempotent type, ψ(εt)Halso is a homogeneous group of idem-
potent type. By Corollary 6.5, we have the isomorphism Gt∼
=ψ(εt)H, whence Ω(G)⊆Ω(H). By
symmetry, we obtain the converse inclusion. Thus, Ω(G)=Ω(H).
Now we can construct an isomorphism from the group Ginto the group Hby the Kaplansky
method. For every type t∈Ω(G), we fix the direct summand Atof rank 1 of the group Gt.Wetake
anonzeroelementat∈Atwith characteristic consisting of the symbols 0 and ∞.Letπt:G→At
be the projection. Then ψ(πt)His a direct summand of rank 1 of the group Ht.Inψ(πt)H,wetake
anonzeroelementbtwith characteristic consisting of the symbols 0 and ∞.Then{at}t∈Ω(G)is a
generator system of the End G-module Gand {bt}t∈Ω(H)is a generator system of the End H-module
H. Any element aof the group Gcan be represented in the form
a=αt1(at1)+αt2(at2)+...+αtk(atk),
where αt1,α
t2,...,α
tk∈End G.Thenweset
ϕ(a)=ψ(αt1)(bt1)+ψ(αt2)(bt2)+...+ψ(αtk)(btk).
Similarly to the proof of Theorems 4.1, 5.1, and 6.3, we can verify that the mapping ϕis well-defined
and it is an isomorphism which induces the isomorphism ψ.
At the end of this section, we use topological isomorphisms to extend the Baer–Kaplansky theorem
(Theorem 5.1) to the case of arbitrary groups.
First, we have the following useful fact. Let Abe a group and εbe an idempotent of the ring EndA.
Then the canonical isomorphism End(εA)∼
=ε(End A)ε, specified in the property (b) of Sec. 2, is
topological if we assume that End(εA) is provided by the finite topology which coincides on ε(End A)ε
with the topology induced by the finite topology of the ring End A.
291
We recall that the torsion part of the mixed group G, i.e., the largest torsion subgroup of G,is
denoted by t(G). In addition, if Gis a torsion group, then t(G)=Gand t(G) = 0 for a torsion-free
group G.
Theorem 6.7 (May [23]).The following assertions hold.
1. Let Gand Hbe t wo groups and ψ:EndG→End Hbe a topological isomorphism. Then there exists
an isomorphism ϕ:t(G)→t(H)such that ψ(η)and ϕηϕ−1coincide on t(H)for every η∈End G.
2. Let Tbe a torsion group and Hbe an arbitrary group. Then every topological isomorphism End T→
End His induced by an isomorphism T→H.
Proof. 1. The required isomorphism ϕcan be constructed by the Kaplansky method which is restricted
here to the torsion parts of the groups Gand H. It is only necessary to clarify the following point.
If εis an idempotent of the ring End Gand εG ∼
=Zpk,k1, then it is clear that ψ(ε)H∼
=Zpk.
Let εG ∼
=Zp∞. Then End(εG)∼
=
Zp(
Zpis the ring or group of p-adic integers) and, consequently,
Endψ(ε)H∼
=
Zp. It follows from this property that ψ(ε)H∼
=Zp∞or ψ(ε)H∼
=
Zp.Thetheorem
that End(εG)andEnd
ψ(ε)Hare topologically isomorphic rings was mentioned before. However, the
finite topologies on the rings End Zp∞and End
Zpare distinct, since the first topology is the p-adic
topology and the second topology is the discrete topology. Therefore, ψ(ε)H∼
=Zp∞is only possible.
Now the way to the application of the Kaplansky method is open.
2. Let ψ:EndT→End Hbe some topological isomorphism. Since ψis continuous, we have that
for an arbitrary element y∈H, there exist elements x1,x
2,...,x
n∈Tsuch that if α∈End Tand
α(xi)=0,i=1,2,...,n;thenwehaveψ(α)(y) = 0. There exists a positive integer msuch that
mxi=0foralli. Hence ψ(m·idT)(y)=(m·idH)(y)=my =0.Consequently,His a torsion group
and we can use Theorem 5.1.
7. Definability of p-groups by Radical of Endomorphism Rings
One can raise the question of the determinability of a group not by the whole ring of endomorphisms
but by some part of it. It follows from Theorem 7.1 that a p-group with an unbounded basic subgroup
is determined by the Jacobson radical of its endomorphism ring (as a ring without unity). Other
similar results are given in the remarks. The section is based on the paper [14]. Various facts about
radicals of endomorphism rings are given in the fourth part of the monograph [17].
Let us make one terminological remark. Group terms applied to a ring, ideal, or module refer to their
additive groups. The same applies to the individual elements of these objects. Thus, for example, the
order of an element of a ring (an ideal or a module) means its order as an element of the corresponding
additive group.
Let Gbe some group. Then J(End G) is the Jacobson radical of its endomorphism ring and K(G)
is the torsion subgroup of the ideal J(End G). It is clear that K(G)isanidealinEndGand the ring
K(G) is not unital . We often write K instead of K(G).
Theorem 7.1. Let Gbe a p-group whose basis subgroup is an unbounded group, and Gan arbitrary
p-group. Then every ring isomorphism ψ:K(G)→K(G)is induced by some group isomorphism
ϕ:G→G, i.e., ψ(η)=ϕηϕ−1for any η∈K(G).
First, we prove a series of auxiliary assertions. But first we note that any ring isomorphism End G→
End G, of course, induces the ring isomorphism K(G)→K(G).
Up to the end of the section, Gis some p-group and N(End G) is the nil-radical of the ring End G,
i.e., the sum of all nil-ideals of End G. We also define the Pierce ideal P(G) of the group G;namely,
we set
P(G)=α∈End G|x∈G[p],h(x)<∞⇒h(x)<h
α(x).
292
It is easy to verify that P(G) is an ideal of the ring End G.Wealsohavetherelation
P(G)=α∈End G|α(pnG)[p]⊆pn+1G[p] for all n0.
Various facts about the Pierce ideal can be found in [17, Sec. 20]. The most important of these is the
inclusion J(End G)⊆P(G). Consequently, we have the inclusions
N(End G)⊆J(End G)⊆P(G).(∗)
Further, we prove that the torsion parts of these ideals coincide. First, we note that if Iis an ideal of
some ring, then t(I) is also an ideal. In addition, t(End G)isap-group in our case.
Lemma 7.2. There exist the relations
tN(End G)=tJ(End G)=tP(G).
Proof. It follows from (∗) that it suffices to show that every element of tP(G)is nilpotent. Let
α∈tP(G).Thenpnα=0forsomen∈Nand αG ⊆G[pn]. Let α=αG[pn+1].Sinceα∈P(G), it
is clear that (α)m=0forsomem∈N.Thenαm+1 =0.
We recall that the ideal K(G) is denoted by the single letter K.
Lemma 7.3. The ideal Kis not bounded if and only if the group Ghas an unbounded basis subgroup.
Proof. We assume that there is a decomposition G=abH,o(a)<o(b). We define an
endomorphism αof the group Gby setting α(b)=aand αaH=0.Thenα∈Kando(α)=o(a).
This proves that the ideal K is not bounded if Ghas an unbounded basis subgroup. To prove the
converse, we assume that G=AD,wherepnA=0forsomenand Dis a divisible group. Let
β∈K. Then pnβ= 0, since a divisible group does not have nonzero endomorphisms of finite order.
Consequently, pnK=0.
We use the following notation. For an endomorphism η∈K and a subset L⊆K, we denote by L·η
the set {λη |λ∈L};thesetη·Lis similarly defined.
Lemma 7.4. Let Gbe a group with unbounded basis subgroup, tbe a positive integer, and let η∈K.
The relation K[pt]·η=0holds if and only if ηG ⊆ptG.
Proof. Let α∈K. Clearly, ptα=0⇔(ptα)G= 0. Therefore, if ηG ⊆ptG,thenK[pt]·η=0.Toprove
the converse, we assume that ηG ptG. We construct an endomorphism λ∈K[pt]withλη =0.We
take an element x∈ηG such that x/∈ptG.Thenh(x)<t. Consequently, there exists a decomposition
G=yY,andx=psy+u,wherepsy=0,s<t,andu∈Y(see Sec. 1). Let o(y)=pnand
m=min{n, t};thenm>s. Since a basis subgroup of the group Gis not bounded, Yhas a direct
summand zof order pkwith k>m. We define an endomorphism λby setting λ(y)=pk−mzand
λY =0.Thenλ∈K[pt]andλ(x)=pk−(m−s)z=0.Consequently,λη = 0, which is required.
Proposition 7.5. If the conditions of Lemma 7.4 hold and η∈K[pt],thenK[pt]·pt−1η=0if and
only if the group Ghas decompositions G=yY=η(y)Xsuch that o(η(y)) = pt.
Proof. First, we assume that K[pt]·pt−1η=0.Then(pt−1η)GptGby Lemma 7.4. Consequently,
there exists a direct summand yof Gwith η(pt−1y)/∈ptG. This implies that the element η(pt−1y)is
of order pand of height t−1. Therefore, η(y)is a direct summand in G(see Sec. 1). It is also clear
that o(η(y)) = pt.
To prove the converse, we assume that the group Ghas the mentioned decompositions. The height
of the element pt−1η(y)islessthant; this implies that (pt−1η)GptG. It follows from Lemma 7.4
that K[pt]·pt−1η=0.
We introduce one new notion.
293
Definition 7.6. ArightK-module Tis said to be strongly homogeneous if Tis a torsion group and
for any nonzero σ, τ ∈T, the intersection σK ∩τK contains an element αwith o(α)=min{o(σ),o(τ)}.
Proposition 7.7. Let the conditions of Lemma 7.4 hold and let η∈K[pt]such that K[pt]·pt−1η=0.
The subgroup ηG is a cyclic group if and only if ηKis a strongly homogeneous K-module.
Proof. By Proposition 7.5, there exist decompositions
G=yY=xX
such that η(y)=xand o(x)=o(η)=pt.
First, we assume that ηG is a cyclic group. Then we have ηG =x. Let endomorphisms τi∈K
satisfy ητi=0,i=1,2. Let o(ητi)=pni.Thennit,i=1,2; without loss of generality, we
assume that n1n2. There exist elements gi∈G,i=1,2 such that ητi(gi)=pt−nix.Wehavethe
decomposition G=zZsuch that the order of the element zexceeds the orders of the elements g1
and g2. We take endomorphisms σi∈K such that σi(z)=giand σiZ=0,i=1,2. We set λi=ητiσi,
i=1,2. Then λiZ=0andλi(z)=pt−nix; therefore, o(λi)=pni=o(ητi), i=1,2. Further, we have
that pn2−n1λ2=λ1and λ1∈(ητ1)K∩(ητ2)K. In addition, the order of the element λ1is equal to the
minimal order of the elements ητ1and ητ2.
To prove the converse, we assume that ηG =x. Then there exists an element gsuch that η(g)=w
for some w/∈x.Ifo(w)=ps, then it is clear that st. Again, we take a decomposition G=zZ
with the property that the order of the element zstrictly exceeds the orders of the elements yand
g. In such the situation, there exist endomorphisms σ, τ ∈K such that σ(z)=y,τ(z)=gand
σZ =0=τZ.Thenwehaveησ(z)=x,ητ(z)=w,o(ησ)=ptand o(ητ)=ps. We assert that the
intersection (ησ)K∩(ητ) K does not contain elements of order ps. We assume the contrary: there exist
α, β ∈K such that the element ησα =ητβ is of order ps.Letabe an arbitrary element of the group G.
We have α(a)=mz +zand β(z)=nz +z for some integers mand nand elements z,z ∈Z.Now
we have ησα(a)=mx,ητβ(a)=nw and mx =nw.Sincew/∈x,wehavethatpdivides n.Then
ps−1ητβ(a)=0andps−1ητβ = 0. The obtained contradiction completes the proof.
We now state the main auxiliary result.
Proposition 7.8. Ap-group Ghas an unbounded basis subgroup if and only if there exists a sequence
1<n
1<n
2<... of positive integers, and there exist elements η1,η
2,... of Ksuch that the following
conditions hold for every i1.
(1) o(ηi)=pni;
(2) K[pni]·pni−1ηi=0;
(3) ηiKis a strongly homogeneous K-module;
(4) the mapping fi:Kηi→Kηi+1,αηi→ αηiηi+1 ,α∈K, is a monomorphism.
In addition, for a given sequence 1<n
1<n
2< ... and given endomorphisms ηi∈Ksatisfying
properties (1)–(4), there exist elements d1,d
2,... in Gsuch that for every i1,wehave
o(di)=pni,η
i(di+1)=di,η
iG=di,di”— is a direct summand of the group G.
Proof. Let Ghave a unbounded basis subgroup. Then for any i1, there exists an element ai∈Gof
order pni,where1<n
1<n
2<..., and the relations
G=aia2...atHt,H
t=at+1Ht+1.
hold. We define endomorphisms ηiby setting ηi(ai+1)=aiand assuming that ηiannihilates all
summands complement to ai+1. It is clear that ηiis of order pni. It follows from Propositions 7.5
and 7.7 that the conditions (2) and (3) hold. The condition (4) is directly verified.
To prove the converse, it suffices to note that by Lemma 7.3, it follows from the condition (1) that
Ghas an unbounded basis subgroup.
294
Now we assume that the conditions (1)–(4) hold and verify the existence of the elements diwith
required properties. By Proposition 7.5, for every i1, there exist decompositions G=yiYi=
xiXisuch that ηi(yi)=xiand o(xi)=pni. It follows from Proposition 7.7 that ηiG=ηi(yi).
This implies the relation G=yiker ηi.Consequently,wehavexi+1 =kiyi+zi,wherekiis an
integer and ηi(zi)=0.
We assume that pdivides ki.Then
pni−1ηiηi+1G=pni−1ηi(xi+1)=pni−1kixi=0.
On the other hand, we have pni−1ηiηi+1 =fi(pni−1ηi)andfiis an injection. This is a contradiction.
Thus, kiis relatively prime to p.
Let d1=x1and let tbe a positive integer such that for all i=1,2,...,t, the element diis already
defined in such a way that di=mixi, where the integer miis relatively prime to pand ηi(di+1)=di
for every i<t. It follows from the relations xt+1 =ktyt+zt(see the above) that ηt(xt+1)=ktxt.
Since ktand mtare relatively prime to p, there exists an integer mt+1 that is also relatively prime
to psuch that the relations
ηt(mt+1xt+1)=mt+1ktxt=mtxt=dt
hold. We set dt+1 =mt+1xt+1.Thus,foreveryi, the group di=xiis a direct summand of the
group Gof order pni, which is required. The remaining properties of the elements diare valid by the
choice of di.
Proof of Theorem 7.1 We assume that the symbol Kdenotes the ideal K(G), and K denotes K(G), as
above. Let ψ:K→Kbe some ring isomorphism of nonunital rings. For an endomorphism α∈K, we
set α=ψ(α)∈K. By Proposition 7.8, there exist positive integers n1<n
2<... and endomorphisms
ηi∈K satisfying the conditions(1)–(4) of Proposition 7.8. The corresponding properties are preserved
under ring isomorphisms, whence the endomorphisms η
i∈Ksatisfy similar conditions (1)–(4). We
apply Proposition 7.8 again and obtain that there exist elements di∈Gand d
i∈Gwith the following
properties. These elements generate direct summands of order pniand ηi(di+1 )=di,ηiG=di,
η
i(d
i+1)=d
i,η
iG=d
i.
We define the mapping ϕ:G→Gas follows. For an element x∈G, we take a positive integer
kwith o(x)<p
nk.Letεbe an endomorphism of the group Gsuch that ε(dk)=xand εannihilates
the complement to dksummand. Then ε∈tP(G)= K by Lemma 7.2. We set ϕ(x)=ε(d
k).
The mapping ϕis well defined. Indeed, we assume that x=ω(dj)forsomejkand ω∈K. Then
dk=(ηkηk+1 ·...·ηj−2ηj−1)dj, which implies (εηk·...·ηj−2ηj−1−ω)dj= 0. Therefore,
(εηk·...·ηj−2ηj−1−ω)ηjηj+1 G=(εηk·...·ηj−2ηj−1−ω)dj=0.
It follows from (4) that (εηk·... ·ηj−2ηj−1−ω)ηj= 0, whence we have εη
k·...·η
j−1η
j=ωη
j.As
a result, we obtain the relation ω(d
j)=ε(d
k) which means that the action of ϕdoes not depend on
the choice of the element dkand endomorphism ε.
Finally, similarly to the proof of Theorem 4.1, it is directly verified that ϕis an isomorphism which
induces the isomorphism ψ.
Remarks 1. It is interesting that a group with a sufficiently rich divisible subgroup is determined
by its topological endomorphism ring in the class of all groups. Namely, May proved (see [22]) the
following assertion: Let a group Gcontain copies of the groups Qand Zp∞for every prime p. Then for
any group H, every topological isomorphism End G→End His induced by some isomorphism G→H
(cf. Theorem 6.7).
There are not many papers devoted to the definability of torsion-free groups by endomorphism
rings. The paper [3] of Bazzoni and Metelli is very important. They proved that a separable torsion-
free group Gis determined by its endomorphism ring in the class of all such groups if and only if every
direct summand of rank 1of the group Gis divided by almost all prime integers. It should be noted
295
that torsion-free groups often have few endomorphisms, and, generally speaking, the endomorphism
ring has little effect on the original group. For torsion-free groups, the isomorphism theorem is a very
rare phenomenon.
For mixed groups, on the contrary, there is a rather rich literature on isomorphism theorems. But
we immediately point out that even for mixed groups Gwith G/t(G)∼
=Q, the following two central
questions have negative answers.
(i) Will the isomorphism G∼
=Hfollow from the isomorphism End G∼
=End H?
(ii) Is every automorphism of the ring End Ginner?
The corresponding examples can be found in the paper of May and Toubassi (see [31]).
If the isomorphism theorem is invalid for some groups, we can extend the problem and try to find
conditions for isomorphism of endomorphism rings of two given groups Gand H. May and Toubassi
(see [35]) did this for mixed groups of rank 1 with totally projective torsion parts.
Various other results connected with the determination of mixed groups by their endomorphism
rings are contained in the papers May and Toubassi [23, 29, 30].
At the end of Sec. 1, we pointed out the close connections of Abelian groups with modules over
domains of discrete valuation. The ring Qpof rational numbers with denominators relatively prime
to pgives an example of such a domain. In fact, the Qp-modules coincide with the groups Gsuch that
nG =Gfor all integers nwith (n, p)=1.
Zpis a complete discrete valuation domain. It is the completion of the ring Qpin the p-adic topology.
The
Zp-modules are also called p-adic modules.
Many papers are devoted to isomorphism theorems for endomorphism rings (or endomorphism
algebras) of mixed modules over discrete valuation domains. It is clear that all the results obtained
in this case are applicable, in particular, to Qp-modules and
Zp-modules. Unfortunately, even if we
accept the strongest assumptions, i.e., if we consider a complete discrete valuation domain Rand
topological isomorphisms of endomorphism R-algebras, even for mixed modules of rank 1, the two
central questions formulated above have a negative solution. This follows, for example, from the paper
of May [28].
Most results on the isomorphism problem for mixed modules refer to modules with totally projective
torsion submodules or to Warfield modules. A typical result here is the following theorem of May and
Tubassi [32]: Let Mbe a mixed m odule of rank 1over a discrete valuation domain Rwith totally
projective torsion submodule. If Nisamoduleofrank1, then every isomorphism EndRM→EndRN
is induced by some isomorphism M→N. It follows from the paper of G¨obel and May [12] that this
result cannot be carried over to mixed modules of other finite ranks, even if one assumes the divisibility
of their torsion-free factor modules. However, this is possible if the domain Ris complete (see [24]).
Other results on this topic can be found in [4–6, 25–27]. Note that very unexpected examples have
been constructed in these and other articles. In particular, they show that there are various serious
obstacles to finding isomorphism theorems for endomorphism rings of mixed modules over discrete
valuation domains, including if they are complete.
As for Theorem 7.1, the papers [13] of Hausen and Johnson and [34] Schulz completely clarify
when for two p-groups Gand H, every ring isomorphism J(End G)→J(End H) is induced by some
isomorphism G→H.
In the papers of Flagg [7–10], the determinability of the module over a discrete valuation domain
by the radical of its endomorphism ring is studied. In [9], the author succeeded in replacing the
endomorphism ring by the radical of the endomorphism ring in well-known theorems on the definability
of mixed modules with totally projective torsion submodules and Warfield modules.
Using some results on isomorphisms of endomorphism rings of modules, one can get acquainted
with the survey [33].
296
Acknowledgment. The work of A. A. Tuganbaev was supported by the Russian Science Foundation
(project No. 16-11-10013).
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P. A. Krylov
National Research Tomsk State University, Tomsk, Russia
E-mail: krylov@math.tsu.ru
A. A. Tuganbaev
Moscow Power Engineering Institute (National Research University), Moscow, Russia;
M. V. Lomonosov Moscow State University, Moscow, Russia
E-mail: tuganbaev@gmail.com
A. V. Tsarev
Moscow Pedagogical State University, Moscow, Russia
E-mail: an-tsarev@yandex.ru
298
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