Representation of rings with faithful polyform modules

Abstract

This article studies subrings which satisfy a density-type criterion called m-density. It is first observed that if V is a faithful quasiinjective R-module then R is an m-dense subring of BiendRV. This is used to obtain the main result which states that R is an m-dense subring of a direct product of rings of linear transformations if and only if R has a faithful locally finite dimensional module whose essential sub-modules are rational.

Full text

COMMUNICATIONS IN ALGEBRA,
14(6), 1141-1169 (1986)
REPRESENTATION OF RINGS
WITH FAITHFUL POLWORM MODULES
J.M. Zelmanowitz
Department of Mathematics
Santa Barbara, California
93106
U.S.A.
ABSTRACT
This article studies subrings which satisfy a
density-type criterion called m-density. It is
first observed that if V is a faithful quasi-
injective R-module then R is an m-dense subring
of BiendRV. This is used to obtain the main
result which states that R is an m-dense subring
of a direct product of rings of linear transforma-
tions if and only if R has a faithful locally
finite dimensional module whose essential sub-
modules are rational.
Introduction
A subring R of a ring T is called an m-dense subring of
T (over the left T-module V) if given any vl,
...,
vk
E
V
and
Copyright
@
1986
by
Marcel Dekker, Inc

1142
ZELMANOWITZ
any t
E
T with tvl,.
. .
,tvk not all
0,
there exist r,s
E
RI
with rtv.
=
sv. for i
=
1,
...,
k and with svl,
...,
svk not all
0.
In
[g]
it was shown that R is isomorphic to an m-dense
subring of a full linear ring (that is, a ring of linear transfor-
mations of a vector space) if and only if R has
a
faithful mono-
form left module. The principal objective of this article is to
obtain a similar description of m-dense subrings of direct pro-
ducts (or finite direct sums) of full linear rings.
The treatment of m-density here, as opposed to
[GI,
is
categorical in spirit and was suggested by
J.
Lambek. The fruits
of this approach appear in
$2,
where it is shown, in particular,
that R is an m-dense subring of BiendRV whenever V is a
faithful quasi-injective left R-module (Theorem
2.4).
This largely
reduces the problem to describing when a ring R has a faithful
quasi-injective left module V with End V a direct product of
R
division rings.
A
facile approach is to insist that V be iso-
morphic to a direct sum iC&Vi with each Vi a monoform R-module
and with distinct Vi "completely incomparable". What is actually
shown is that if
M
is a faithful left R-module whose essential
submodules are rational and which is the sum of its finite dimen-
sional submodules then
fi,
the quasi-injective hull of
N,
con-
tains a faithful quasi-injective submodule V of the above type
(Proposition
4.2).
The converse, that an m-dense subring of a
direct product of full linear rings has a faithful module of this
type, is relatively easy to show. One attractive consequence
(Corollary
4.4)
is that a ring is isomorphic to an m-dense subring
of a finite direct sum of full linear rings if and only if it has a

RINGS WITH FAITHFUL POLYFORM MODULES
1143
faithful finite dimensional module whose essential submodules are
rational.
The organizational plan of this paper is to summarize the
necessary background on quasi-injective modules in $1, to treat
m-density in
$2,
to study modules whose essential submodules are
rational in
$3,
to give the main results in
$4,
and to examine the
behavior of these properties under certain ring constructions in
$5.
$1. Preliminaries
We collect here the various conventions, definitions and basic
facts which will be used in subsequent sections. The reader famil-
iar with this topic is advised to go directly to
$4,
referring back
to earlier sections as necessary.
It is important to note that, unless otherwise specified, a
module M is a left module over a ring R which does not neces-
sarily possess an identity element. Homomorphisms will consistent-
ly be written on the side of a module opposite to that of the
scalars. For module theory in this generality,
[&]
and
[g]
are
basic references.
Sometimes, to compensate for the absence of an identity
element, we will limit our attention to cofaithful modules, where
M
is called cofaithful if
Rm
#
0
whenever
0
#
m E M.
R'
will
denote the ring with identity containing R obtained by the
customary construction, so that for m E M, ~lm
=
(rm
+
kmlr E R,
kE
4.

1144
ZELMANOWITZ
A partial homomorphism from
M
to an R-module
N
is an
element f
E
HomR(N1,N) with Mt an R-submodule of
M.
When
M
=
N, f is called a partial endomorphism of
M.
A monoform
module is one in which all non-zero partial endomorphisms are
monomorphisms.
A
monoform module is uniform; that is, any two
non-zero submodules have non-zero intersection.
M is a quasi-injective module if every partial endomorphism
of M extends to an endomorphism of M; equivalently, if M is a
fully invariant submodule of its injective hull. A module M can
always be embedded in a smallest quasi-injective module, called its
quasi-injective hull and denoted by
fi,
which is unique up to iso-
morphism over M. We will always write E(M) for the injective
hull of
M.
It is a fact that
fi
E
MA
where
A
=
EndRE(M).
Hence, when an injective hull E(M) has been specified, we will
always assume that M
5
fi
5
E(M). Also, it is now readily deduced
that
fi
=
MA where A
=
~nd~fi. Proofs of these facts can be found
in
[4]
-
and
[6].
-
A submodule N of M is called rational (in M) if
HomR(M/N,
g)
=
0;
equivalently, if for each m
E
M and
0
#
mt
E
M there exists r
E
R1 with rm
E
N and rm'
#
0
[4;
Proposition
7.21.
The following conditions on a non-zero module
M
are equivalent: (i) M is monoform; (ii) every non-zero submodule
of M is rational; (iii) ~nd~fi is a division ring
[?;
Proposi-
tion
7.51.
We will need the following elementary facts about quasi-
injective modules.

RINGS WITH FAITHFUL POLYFORM MODULES
1145
LEMMA 1.1. Suppose that M
=
.@
M. If, for each i
E
I,
161
1'
Mi is quasi-injective and the family {Mi/i
E
I]
is completely
incomparable, then M is quasi-injective.
Here a family {M. li
E
I) of R-modules is called completely
incomparable if there are no non-zero partial homomorphisms from
Mi to M. whenever i,
j
E
I and i
#
j
.
J
Proof.
It suffices by
[I;
Proposition 16.131 to show that M is
M.-injective for each
i
E
I. This is a standard argument using
the incomparability hypothesis.
D
LEMMA 1.2. Suppose that M1
and
M2
are
R-submodules of E(M)
with E(M1) and E(M2) contained in E(M).
If
E(M1)
Z
E(M2)
Proof. Let f
E
HomR(E(M1), E(M2)) be the assumed isomorphism and
-
extend it and its inverse to R-endomorphisms of E(M). Denote
these extensions by f and g, respectively. Then
(E(M~)
ri
fi)fc
-
E(M~)~
n
fifz
E(M~)
n
fi
and
(E(M~)
n
fi)f-I
=
(E(M2)
n
fi)g
c
-
E(M~)~-'
li
fig
c
E(M
)
fI
fi
because
fi
is a fully
-
1
invariant submodule
of
E(M). Hence
f
restricts to an isomor-
phism of
E(M~)
n
fi
onto
E(M~)
n
17.
LEMMA
1.3.
Suppose that E(M)
=
1
Ei is an (internal) direct
i
E
I

1146
ZELMANOWITZ
sum of submodules Ei,
i
I.
Then, for any subset
J
of
I,
Proof.
2
is clear. On the other hand, for each
j
E
J,
let
-
-
Pj
:
E(M)
+
Ej denote the canonical projection onto E.. Then
3
fip.
c
E.
n
fi
because
fi
is fully invariant in E(M).
It
follows
J-
J
that
(
,Z
E~)
il
ic
-
,Z
(E.
nri).
j€J j€~
$2.
A
general density theory
The approach to density that is given below is not new. The
essential ideas or variations of them have appeared regularly in
the literature (for instance, in
15)
and
[?I).
For
M
an R-module and
A
an index set, PI(*) and
M
A
will
denote, respectively, the direct sum and direct product. We
regard M(*) as an R-submodule of
M~
and write a typical
element of
M*
as (mala with each
ma
€
M.
An R-submodule
L
of
MA
will
be called Biend M-invariant if given any element
R
lmaL
A
in
L
and any
t
E
BiendRM, the element {tmalcr
A
is
in L. We can provide sufficient criteria for such invariance to
occur.
PROPOSITION
2.1.
Suppose that L
is
R-submodule of
M*
such
that either (i)
L
is generated by
M
(ii)
L
c
-
M(*)
,nd
M(~)/L is cogenerated by
M.
Then
L Biend M-invariant.
R

RINGS
WITH
FAITHFUL POLYFORM MODULES
Proof. Assume that (i) holds and choose an R-epimorphism
:
M'~)
+
L for some index set
B.
Let (majaEA be an arbi-
trary element of L and let
t
be arbitrary element of BiendRM.
For each a
E
A
and
f3
E
B consider the sequence
where
rg
is the canonical embedding of
M
into the
@-coordinate of M(~), na is the canonical epimorphism of
M~
onto the a-coordinate, and
E
is
the inclusion homomorphism.
It
suffices to show that (tma)aE
A
is in the image of
E.
k
which
is
in the image of
r.
The situation when (ii) holds
is
treated with a dual proof.
0
Let R be a subring of a ring
T
and let
M
be a left
T-module. We say that
R
is
a dense subring
of
T
(over
M)
if
given any
t
E
T
and
ml,.
.
.
,mk
E
M
there exists
s
E
R1
with
tm.
=
sm for all
i
=
1,
...,
k.
More generally, adopting the
i
language of
[g],
we
will
say that R
is
an m-dense subring of
T
(over
M) if given any
t
E
T and
ml,
...,
mk
E
M
with

1148
ZELMANOWITZ
tml,.
.
.
,tmk not all 0, there exist r,s
E
R1 with rtm.
=
sm.
for all i
=
1,
...,
k, and with sml,
...,
smk not all
0.
We will be most interested in the special case when M is a
left R-module and T
=
BiendRM. In this situation, setting
iR(M)
=
{r
E
R jr~
=
03, R/iR(M) is naturally isomorphic to a sub-
ring of T; and we will therefore always regard R/%(M) as a sub-
ring of
T.
The essence of the Chevalley-Jacobson density theorem for
simple or semisimple modules is contained in the following conse-
quence of the proposition.
COROLLARY
2.2.
Suppose that M is an R-module with the property
that for each positive integer k and for each R-submodule L
of M(~), either L is generated by M M(~)/L is cogener-
-
ated by
f.3.
Then
R/gR(M) is a dense subring of BiendRM.
Proof.
Given t
E
BiendRM and ml,
...,
mk
E
M,
set L
=
~'(m~,
. .
.
,mk)
5
M(~)
and apply the preceding proposition.
Observe that when M is a semisimple R-module,
(ml,.
. .
,mk)
E
R(ml,.
.
.
,mk)
,
so we may choose L
=
R(ml,.
.
.
,mk)
in the above proof and get the usual density statement that given
t
E
BiendRM there exists s
E
R (rather than s
E
R')
with
tm.
=
sm. for i
=
1,
...,
k.
The m-density condition that will be required later in this
article will be deduced from the next corollary.

RINGS
WITH
FAITHFUL POLYFORM MODULES
1149
COROLLARY 2.3 Suppose that M is an R-module with the property
that for each positive integer k and each (cyclic) R-submodule
K
or
M(')
there exists an essential extension L
of
K in
-
M(~)
with either L generated by M
M(~)/L
cogenerated by
M. Then R/%(M) is an m-dense subring of BiendRM.
Proof. Given t
E
BiendRM and ml,
...,
mk
E
M with tml,
...,
tmk
not all 0, set
K
=
R1(ml,.
.
.,mk)
5
M(~)
and let
L
be an
essential extension of
K
in
M(~)
satisfying the stated hypo-
thesis. By the proposition, 0
#
(tml,
...,
tmk)
€
L,
so there
exist r,s
E
R1 with 0
#
r(tm ,tmk)
=
s(ml,.
..
,mk)
E
K.
0
THEOREM
2.4.
Lf
M is a quasi-injective R-module then R/%(M)
is an m-dense subring of BiendRM.
Proof. For every k
2
1, V
=
M(~)
is quasi-injective
[I;
p.1911.
For K any R-submodule of V, we may decompose E(V)
=
K'
0
E'
with K an essential submodule of
K'.
From Lemma 1.3 we know
that
V
=
(Kt
n
V)
CE
(E'
fI
V),
and so
K'
V is generated by M.
Hence we can apply the preceding corollary with L
=
K'
n
V to get
the desired conclusion.
0
Part of the m-density theorem for rings with a faithful and
cofaithful monoform R-module M states the following
[g;
Pro-
positions 2.1 and 2.51
.
Given any t
€
Biend
fi
and any
R
-
ml,
...,
mk M with tml,
...,
tmk not all
0,
there exist r,s
E
R

1150
ZELMANOWITZ
(not merely in
R1)
with rtm.
=
sm. for all i
=
1,.
.
.
,k,
and
with sml,.,.,smk not all
0.
This is easily deduced from the
preceding theorem by applying the fact that
3
is cofaithful
whenever
M
is.
$3.
Modules with essential submodules rational
A
natural generalization of a monoform module is a module
whose partial endomorphisms have c-losed kernels; that is to say,
if f is a partial endomorphism then kernel f has no proper
essential extension in the domain of f. The next proposition
provides alternative descriptions of such modules.
PROPOSITION
3.1.
The following conditions are equivalent for an
(i)
All partial endomorphisms of
M
have closed kernels.
(ii) Every essential R-submodule of
M
is rational
(iii) End
fi
is a (von Neumann) regular ring, where
fi
denotes
R
the quasi-injective hull of M.
Modules satisfying condition (ii) played
a
prominent role also
in
[g;
p,
1472
ff], where the implication (ii)
=>
(iii) was
demonstrated. It seems appropriate to call a module which satis-
fies the conditions of this proposition a polyform module. For
example, any nonsingular module is polyform.

RINGS WITH FAITHFUL POLYFORM MODULES
1151
Proof. (i)
=>
(ii) Let N be an essential R-submodule of
N
-
and let m E M,
0
#
m'
E
M
be given. We must show that
(N:m)lmt
#
0
where (N:m)'
=
{r ER11 rm
CN).
Without loss of
generality, we may assume that m
#
0.
If, to the contrary, (N:m)'m'
=
0, then f
:
R1m
+
M
defined by (rm)f
=
rm' for r E R1 is a well-defined non-zero
partial endomorphism of
M
because (0:m)l
c
(~:rn)l
c
(0:m')l.
- -
Also, (N:m)'m
=
N
fl
R1m is an essential R-submodule of R1m
because
N
is essential in
M.
But (N:m)lm
c
kernel f, so by
-
hypothesis f
=
0, a contradiction.
i
=>
ii It suffices to show that kernel a is closed in
fi
for every a ~nd~fi. For then kernel a is
a
direct summand of
, and it readily follows (using the fact that
fi
is quasi-
injective) that End
fi
is a regular ring.
R
Let a
E
EndRfi be given and let K be a maximal essential
extension of kernel a in fi.
K
is a direct summand of
fi
[A;
Corollary
3.31.
Let a'
E
EndRM be the extension of
@IK
defined
by setting a'
=
0
on a complementary summand of
K
in
fi.
Then
kernel a' is essential in fi. It follows that, for any
@
E A
=
EndRfi, kernel @at
il
M
=
(kernel at)@-I
I7
M
is
an essential sub-
module of M.
By
hypothesis,
0
=
HomR(M/kernel pa'
n
M,
fi)
;
HomR(M@a'
,
M),
so
we
conclude that
Mpal
=
0.
Since
@
E
A was
arbitrary, Ma'
=
Ma'
= 2
MPa'
=
0.
Thus a'
=
0
and
so
PEA
kernel a
=
K, a direct summand of M.
(1)
=
(i) If, to the contrary, there exists a non-zero
partial endomorphism of M with kernel not closed, then we may

1152
ZELMANOWITZ
construct
0
f
a
E
HomR(N,M) with N an essential R-sumodule of
M
and kernel
a
essential in N. Extend
a
to
a'
E
~nd~fi;
kernel
a'
is then an essential R-submodule of fi. Since ~nd$
is regular, kernel
a'
is also a direct summand of fi. Hence
a'
=
0,
a contradiction.
We remark that if
M
satisfies the conditions of the preced-
ing
proposition then M is a dimension module in the sense of
[?I;
in particular, a sum of two finite dimensional R-submodules of M
is again finite dimensional
[?;
Proposition
I].
In view of this
fact,
it is evident that the following three conditions are equiv-
alent for a polyform module
M.
(i) Each finitely generated submodule of M is finite
dimensional.
(ii) Each cyclic submodule of
M
is finite dimensional.
(iii)
M
is the (not necessarily direct) sum of its finite
dimensional submodules.
It seems appropriate to call a modu
locally finite dimensional (1. f. d.)
is trivally 1.f.d.
.le which satisfies condition (i)
.
A
finite dimensional module
We will also need to know that, under certain conditions, a
direct sum of polyform modules is a polyform module. For this we
need a preliminary step.
LEMMA
3.2.
[2;
-
Lemma
151
Suppose that V1,
...,
Vn,
U
is a family
of R-modules such that partial homomorphisms between distinct V.
-

RINGS
WITH FAITHFUL POLYFORM MODULES
1153
and partial homomorphisms from any Vi to
U
have closed kernels.
-
n
--
Then partial homomorphisms from
O
Vi
to
U
have closed
i=l
kernels
PROPOSITION
3.3.
Let
V
=
@
Vi be a direct sum of R-modules.
iEI
If
partial homomorphisms from Vi
to
V.
have closed kernels for
J
every
pair
i,
j E
I,
then
V is a polyform module.
Proof. Let f be a partial endomorphism of
V
and let
K
be any
-
essential extension of kernel f in the domain of f. It suffices
to prove that
K
=
kernel f.
Given x
E
K
there exists a finite subset F
c
-
I
such that
x is an element of V
=
0
Vi. Let f' denote the restriction
i€F
of f to VFfl domain f; and, for each i E I, let p.
:
VF
+
Vi
1
denote the projection homomorphism. Then ftpi is a partial homo-
morphism of VF into Vi. Since V
n
kernel f
=
kernel f'
5
F
kernel ftpi, kernel f'pi is an essential submodule of
VF
n
K.
By the previous lemma, f'p.
=
0
for each i
E
I. Hence
xf
= xf' =
0, and because x
€
K
was arbitrary we conclude that
K
=
kernel f.
0
$4.
Rings with faithful polyform modules
We
will henceforth be concerned with locally finite dimension-
al polyform modules. In view of the remark following Proposition

1154
ZELMANOWITZ
3.1,
a polyform module
M
is
1.f.d. if and only if each element of
M
lies in a finite dimensional submodule of
M.
To describe m-dense subrings of direct products of full lin-
ear rings we need the following elementary observations. If
{Vili E
I]
is
a family of right modules
Vi
over rings
Ai
and
A
=
n
A,,
then
V
=
O
Vi
is a right A-module under the
iEI
1
E
I
action defined by v.
.
. {Giji
I
=
{viGili
I
for
1
lE1
vi
€
Vi,
bi
E
Ai.
Furthermore, End
V
ll
End(VilAi
A
-
iE1
under the ring homomorphism which ,naps
t
E
End
VA
to {ti)i
I,
where
t.
=
tlV. can be regarded as an element of End(Vi)Ai;
and each
Vi
is in a natural way an End VA-submodule of
V
under
the action defined by t-v.
=
t.v. for
t
E
End
VA
and vi E
Vi.
1
11
If there exist division rings
Ai,
i
E
I, and right
Ai-
vector spaces
Vi,
i
E
I,
such that
R
is an m-dense subring of
End
VA
over
V
where
A
=
ll
A. and
V
=
O
V.
then (iden-
~EI'
l€I
1'
tifying End
VA
with
ll
End(Vi)Ai) we
will
say that
R
is
an
i
€1
m-dense subring of the direct product of full linear rings
We can now state the main result of this paper. Its proof
will
occupy the rest of this section.
THEOREM
4.1.
A
ring
R
has a faithful locally finite dimensional
polyform module and only if
it
is
isomorphic to an m-dense subring
ps
.
A
key step in the proof is to show that starting from a faith-
ful 1.f.d. polyform module one can construct a faithful polyform

RINGS WITH FAITHFUL POLYFORM MODULES
1155
module whose endomorphism ring is a direct product of division
rings. This is accomplished in the next proposition.
PROPOSITION
4.2.
Suppose that M is an 1.f.d. polyform
R-
module. Then there exists a completely incomparable family
{Vili
E
I)
of quasi-injective monoform R-submodules of the quasi-
injective hull
r?
of
M such that
2
V. is a direct sum in
fi
iEI
and
!2
(
1
Vi)
=
QR(M).
-
iEI
Proof. One begins by applying Zorn's lemma to choose a maximal
completely incomparable family (Mi li
E
I] of uniform submodules
Mi of M. Since M is polyform, each Mi is monoform and every
uniform submodule of M is subisomorphic to precisely one Mi
from this family.
We first show that the sum
1
Mi is direct. For if not, let
i €1
m.
+
---
+
m.
=
0
with each
0
f
mi.
E
Mi, and i l,...,ik dis-
l1
Ik
J
J
tinct in
I
be a dependent sum of shortest length k
>
1.
Observe
that for each r
E
R~,
rm.
=
0
if and only if rm.
=
0.
Hence
l1
Ik
the assignment rm. rm. defines
a
non-zero partial
I1
Ik
homomorphism from Mi to Mi
,
a contradiction.
1
k
Fix an injective hull E(M) of M and a quasi-injective hull
fi
with M
5
fi
5
E(M). For each i
E
I
choose an injective hull
E(Mi
)
5
E(M) and set Vi
=
E(Mi)
n
r?.
The sum
2
Vi is direct
i
GI
-
--
because each
M.
is an essential submodule of V. and the sum
1
Mi is direct. It is easy to see that each Vi is a fully
i
E
I

1156
ZELMANOWITZ
invariant submodule of E(M.)
=
E(Vi) and hence is quasi-
injective. Each Vi is monoform because it is a uniform
submodule of fi. Also, the family {Vi(i
E
I]
is completely incom-
parable because the family {Mi i E I] is completely incomparable
and each Vi
5
fi.
Set V
=
2
Vi. It remains only to prove that QR(V)
=
%(M).
iEI
Since
fi
=
MA
where
A
=
~nd~fi and V
5
fi, we know that kR(M)
=
iR(fi)
5
%(V). For the reverse inclusion, let m
E
M be arbi-
trary. It suffices to show that gR(V)
5
%(m).
Since
M
is 1.f.d. there exists a finite dimensional
R-
submodule N of
M
with mEN. We may choose uniform submodules
k
k
N1,
...,
Nk of N such that
2
Ni is direct and
@
E(Ni)
=
i=l i=l-
E(N)
5
E
(M)
.
Applying Lemma 1.3, we have
m
E
E(N)
n
M
=
k
@
(E(Ni) flfi). Each Ni, for i
=
1
,...,
k, is subisomorphic
i=l
to some M. with ji
E
I, and so we can conclude from Lemma 1.2
J
..
I
that E(Ni)
nfi
E
E(M.
)
nr;i
=
V.
.
Hence gR(V)
=
k
J
i
J
i
n
kR(vi)
5
n
Q~(E(N~)
n
B)
=
Q~(E(N)
n
8)
c
-
%(m>. This
CO~-
iC1 i=l
completes the proof.
For the proof of the converse in Theorem
4.1
we utilize the
next result.
PROPOSITION
4.3.
-
If
R
is an m-dense subring of a direct product
over
V
=
@
V.
with each Ai a division ring then
inE
fnd(vi)~i
-
i
II
{Vi(i
E
I]
is a completely incomparable family of monoform R-
modules.

RINGS WITH FAITHFUL POLYFORM MODULES
1157
@.
Here we are identifying
ll
End(Vi)Ai with End
VA
where
iEI
A
=
ll
Ai,
and the R-module structure of each
Vi
is
induced
iE1
from
its
End VA-module structure as described above.
Let f
E
HomR(U,V.) be a non-zero partial homomorphism from
J
Vi
to
V.,
i,
E
I.
Choose u
E
U
with uf
#
0. If uf
$?
uA
J
then there exists
t
E
End
VA
with tu
=
0 but t(uf)
#
0. We
apply m-density to choose
r,s
E
R1 with rtu = su and rt(uf)
=
s(uf)
#
0. Then su = 0 and so s(uf)
=
(su)f
=
0, a contradic-
tion. So
it
must be the case that uf E uA. In particular then,
V.
=
V.
and we have shown that {Vi
(i
E I) is a completely incom-
13
parable family of R-modules.
It
remains to prove that f is a monomorphism. To see this,
let 0
#
v E
U
be arbitrary. Since uf
E
uA, we can write
uf
=
ud for some 0
#
d
E
Ai5
A. Choose
t
E
End
VA
with
tu
=
v, and apply m-density to choose
r,s
E
R~
with rtu
=
su# 0. Then suEVi, so 0
#
(su)d= s(ud)
=
s(uf)
=
(su)f
=
(rtu)f
=
(rv)f
=
r(vf), and hence vf
#
0. Since 0
#
v E
U
was
arbitrary, f
is
a monomorphism.
13
Proof of Theorem 4.1. Assume that
M
is
a faithful 1.f.d. poly-
form R-module. From Proposition 4.2 there exists a completely
incomparable family {V.
1
i
E I) of quasi-injective monoform R-
submodules of
fi
such that
2
V.
is
a direct sum and
V
=
2
V
iE1l
~EI~
is
a faithful R-module. Furthermore,
V
is quasi-injective by
Lemma 1.1. Hence, by Theorem 2.4, we may regard R as an m-dense
subring of End
VA
over
V,
where
A
=
EndRV.

1158
ZELMANOWITZ
Set Ai
=
EndRVi for each i
E
I.
Then each Ai is a divis-
ion ring (because V. is quasi-injective and monoform) and there
is a natural isomorphism A
E
Il
A, (because the family
iCI
(Vi li
C
I) is completely incomparable). Hence End VA
Z
n
End(Vi)Ai, a direct product of full linear rings
iC1
Conversely, assume that
R
is an m-dense subring of a direct
product
n
End(Vi)Ai of full linear rings (over V
=
@
Vi).
iEI
iEI
-
-
From Propositions 3.3 and 4.3, we know that each Vi is a uniform
R-module and that V
=
@
Vi is a faithful R-module whose par-
i€I
tial endomorphisms have closed kernels. Thus V is
a
faithful
1.f.d. polyform R-module.
o
We have the following immediate corollaries of the proof of
the theorem.
COROLLARY 4.4.
A
ring
R
has a faithful finite dimensional poly-
form module if and only if R is isomorphic to an m-dense sub-
ring of a finite direct sum of full linear rings.
COROLLARY
4.5.
[12;
-
Theorem 2.31
A
ring R has a faithful mono-
form module if and only if R is isomorphic to an m-dense subring
of a full linear ring.
In fact, the definition of m-density can be strengthened
somewhat when one is dealing with
a
faithful monoform module; see
&I.

RINGS
WITH
FAITHFUL POLYFORM MODULES
1159
By strengthening the faithful hypothesis we can also special-
ize Corollary
4.4
to obtain
a
characterization of rings whose
maximal left quotient rings are semisimple artinian. Another
description appears in
[G;
Corollary
5.41.
THEOREM
4.6
A ring R has a maximal left quotient ring which is
semisimple artinian if and only if there exists a cofaithful finite
dimensional polyform R-module
M
such that RR(ml,.
. .
,mk)
=
0
for some ml,
...,
mk
E
M.
Proof. Assume that
M
is a cofaithful finite dimensional polyform
-
R-module with %(ml,.
.
.
,mk)
=
0
for some ml,.
.
.
,mk
E
M.
We can
repeat the construction employed in the proof of Proposition
4.2
to
obtain a decomposition
fi
=
V
@
. .
@
Vm
@
Vm+l.
@
'
@
Vn where
1
each V. is a quasi-injective monoform R-module, the family
1
(V1,.
.
.
,VmJ is completely incomparable, and each V. for
J
j
=
m+l
,...,
n is isomorphic to precisely one V. for some
Set V
=
V1@
-..@
Vm and W=Vm+l@
--*@
Vn. Then V is
a faithful quasi-injective module (Lemma 1.1) and there exists a
positive integer q and an R-epimorphiam
$
E
H~~~(v(~),w), where
v(~'
is the direct sum of q copies of V. For each
i
=
lk, write mi
=
v.
+
w. with vi
E
V and wi
E
W; and,
11
in turn, write each
w. =
(u. ll,.-- ,u.
)$
where every
u..
E
V.
14
1J
Then gR(vl,.
.
.
,vk,ull,.
. .
,ulq,.
.
.
,ukl,.
. .
,ukq)
5
%(ml,.
.
.
,mk)
=
0.
Hence by a well-known lemma (see
[g;
1.1]), V is a finitely

1160
ZELMANOWITZ
generated module over EndRV. Now End V is a finite direct pro-
R
duct of fields, so Biend V is semisimple artinian. Also, R is
R
m-dense in BiendRV by Theorem
2.4.
From this information one
readily concludes that BiendRV is the maximal left quotient ring
of R.
For the converse, we refer the reader to
[g]
where it is
shown that if R has a maximal left quotient ring which is semi-
simple artinian then
R
has a faithful finite dimensional nonsin-
gular module
M
with gR(ml,
...,
mk)
=
0
for some ml,
...,
mk
E
M.D
A commutative ring has a faithful monoform module if and only
if it is a domain. This is because the annihilator of such a mono-
form module is a prime ideal. More generally, we have the follow-
ing results for commutative rings with faithful polyform modules.
THEOREM
4.7.
Let
R
be a commutative ring.
(1)
R
has a faithful (locally finite dimensional) polyform
module if and only if R is semiprime.
(2)
R has a faithful finite dimensional polyform module if
and only if the total quotient ring of
R
is a finite direct
product of fields.
Proof.
(1) Assume that M is a faithful polyform R-module. For
each
0
#
a
E
R let
2
E
EndRM be multiplication by a. Then
for each
kernel
sk
is an essential R-submodule of kernel a

RINGS
WITH
FAITHFUL POLYFORM MODULES
1161
positive integer k. For, if m E kernel
ik+l
is given then
k
k
either a m
=
0
and m E kernel
ik
or else a m
#
0
and so
0
#
k
Ak
a m
E
kernel a
.
Since
M
is polyform, kernel ak
=
kernel a
k+ 1
for every positive integer k. From this it follows that the ann
hilator of
M
is a semiprime ideal of R. Hence
R
is semiprime
Conversely, if
R
is semiprime then
n
Pi
=
0
where
iEI
{Pi
li
E
I]
is the set of minimal prime ideals of R. Then
{R/Pi li
E
I] is a completely incomparable family of monoform
R-modules. Hence, by Proposition 3.3, V
=
'3
R/P. is a faithful
~EI
1.f.d. polyform R-module.
(2)
Let
M
be a faithful finite dimensional polyform
R-module. From Proposition
4.2
we know that there exists
a
finite
completely incomparable family {VI,.
.
.
,V of monoform R-modules
k
such that V
=
V1@
'3
Vk is a faithful R-module. For each
i
=
,,k set
Pi
=
RR(Vi). Each
Pi
is a prime ideal and
k
n
Pi
=
0.
It is then easily seen that the total quotient ring of
i=l
R is a finite direct product of fields.
Conversely, if the total quotient ring of R is a finite
product of fields then
R
has finitely many minimal prime ideals
and their intersection is
0.
Let PI,
...,
Pk
be the distinct
minimal prime ideals of R. Then, as in (1) above,
V
=
R/P1@
@
R/Pk is a faithful finite dimensional polyform
R-module.
$5.
Related rings
The principal objectives of this section are to show that the
property of being an m-dense subring of a direct product (or

1162
ZELMANOWITZ
finite direct sum) of full linear rings is a Morita invariant of
rings with identity element and is inherited by polynomial rings.
Let
M
be an R-module and (R, RPS, SQR, S) a Morita con-
text with inner products
(
,
)
:
PXQ
+
R and
[
,
]
:
QxP
+
S.
-
For each p E P, we define p
E
Hom (Q @3RM, M) on generators of
QaRM by
p(q@
m)
=
(p,q)m for q E Q, m
E
M. Set K
=
2
kernel 5. Then
K
is a left S-submodule of Q BRM and
Q
@RM
N
=
------
is a left S-module. Observe that for any
q
E
Q,
p
E
P
and x
E
Q QRM,
4
8
F(x)
=
[q,p]x. For
L
any subset of
I,
- -
Q GRM, we set P(L)
=
{
2
pi(xi)
I
pi
E
P,
xi EL), an R-submodule
i=l
-
-
of M. The results below refer to this setting
LEMMA
5.1.
Assume that (P,Q)m
#
0
whenever
0
#
m
E
M.
(i)
If
RM is cofaithful then
SN
is cofaithful.
(ii) If
K
5
Lo
5
L
are
S-submodules of Q BRM such that
-
Lo/K is essential in L/K
-
then P(Lo) is an essential
R-submodule of F(L).
(iii)
If
{Li /i
E
1) is a family of S-submodules of
Q
BRM
Li
+
K
@,"
such that
t
-
is a direct sum in
N
=
-
iEI
K
then
2
F(L~)
is a direct sum in M.
i EI
Proof. (i) is easy to verify
[e;
Proposition
4.21.
We adopt the convention that for any subset Mo of M,
k
Q
8
Mo denotes
(
t
qi
@
mi
E
Q
8
RM
Iqi
E
Q, mi
E
Mo)
i=l
(ii) We must show that for each non-zero R-submodule Mo of

RINGS WITH FAITHFUL POLYFORM MODULES
1163
-
,-,
P(L), Mo
n
F(Lo)
#
0. By assumption then P(Q
@
Mo)
=
(B,Q)M~
#
0,
so Q
8
Mo K. Since Lo/K is an essential submodule of L/K,
((Q
@
Mo)
+
K)
n
Lo
g
K. Hence 0
#
F(((Q
8
MJ
+
K)
n
Lo)
5
F(Q
s
M~)
n
F(L~)
=
(P,Q)M~
n
F(L~)
5
M~
n
Fao).
-
(iii) We show that for each j E I, P(Lj)
n
I
F(L~)
=
0.
i#
j
N
we apply the assumption to conclude that P(L.)
n
I
?(Li)
=
0.
0
J
i#j
PROPOSITION
5.2
Assume that (P,Q)m
#
0
whenever 0
#
m E
M.
(i)
If
L is an S-submodule of Q
O
M
which contains
K
R
then dim L/K
$
dimRF(~),
where
dim denotes uniform dimension.
S
Q
@$
(ii)
-
If M is a polyform R-module then N
=
-
is
a
K
-
polyform S-module.
Proof. (i) Suppose that we have a family (L.
1
i
E
I] of S-
submodules of QBRM with each K
3
Li
C_
L and
1
Li/K a direct
ic
I
sum. From part (iii) of the preceding lemma we know that
1
F(L~)
iE I
is a direct sum of non-zero submodules of F(L). This proves the
desired inequality.

1164
ZELMANOWITZ
Q
BRM with
(ii) Let g be a partial endomorphism of
-
domain
g
=
L/K and kernel g
=
Lo/K, where K FLo
5
L are
S-
submodules of
Q
BRM. Define f:
P(L)
+
M as follows. For
pi
E
P and xi
E
L (1
.
k), choose yi
E
QBRM
SO
that
k
w
k
,./
(xi
+
K)g
=
yi
+
K, and set
(
1
pi(xi))f
=
2
pi(yi). It is
i=l i=l
clear that f will be a partial endomorphism of RM if it is
well-defined. We will show somewhat more; namely, if
k kN
2
pi(xi)
E
Pi(~o)
then
1
pi(yi)
=
0.
From this it will
i= 1 i=l
,-
follow that f is well-defined and that P(Lo)
c
-
kernel f.
k%
Set
z
=
2
pi(xi)
E
B(L~). For any
q
E
Q,
i= 1
q8
z
E
QO
F(L~)
=
[Q,P]Loc Lo. Hence, in
N,
k
It follows that
Q
@
2
si(Yi)
C
-
K and hence
i=l
k
w
k
,..
0
=
P(Q
8
2
pi(yi))
=
(P,Q)
2
pi(yi). By assumption then,
i=l i=
1
In order to prove (ii), it suffices to show that if Lo/K is
an essential S-submodule
of
L/K then g
=
0.
To see this,
observe first that by the preceding lemma
F(L~)
is then an

RINGS
WITH FAITHFUL POLYFORM MODULES
1165
esseritial R-submodule of
F(L).
Since partial endomorphisms of
RM have closed kernels, f
=
0.
This means in particular that
-
P(y)
=
0
for each coset y+K in the image of g. So g
=
0.
a
THEOREM
5.3.
Suppose that (R,
RPS,
SQR, S) is a Morita context
which satisfies (Ps
,Q)
#
0
and
(P,Q)m
#
0
whenever
0
#
s
E
S,
0
#
m E M.
-
If M is a faithful (locally) finite dimensional
QeRM
polyform left R-module then N
=
-
is a faithful (locally)
finite dimensional polyform left S-module, where
k k
K=
{
t
qi8milqi E
Q,
mi€
M,
and
t
(p,qi)mi
=O
i= 1 i=l
for every p
E
P).
Proof. The hypothesis that (Ps,Q)
#
0
whenever
0
#
s E S
guarantees that SN is faithful. This is an easy calculation
[3;
-
Theorem
4.21.
In view of the preceding
only observe that if M
=
2
Mi with each
iE1
Q
@$
sional R-submodule of
M,
then N
=
-
K
(Q
@
Mi)
+
K
and dimS
K
6
dimR P(Q
8
Mi)
=
dimR(P,Q)Mi
2
dimRMi
<
for each iE
I.
In particular, SN will be finite
dimensional when RM is.
proposition we need
M. a finite dimen-
1
(Q
@
Mi)
+
K
=t
iE1 K
This theorem shows that possessing a faithful (locally) finite
dimensional polyform module is a Morita invariant of rings with
identity elements. More generally, we have the following
consequences.

1166
ZELMANOWITZ
COROLLARY 5.4. (1)
If
rR(R)
=
{r
E
R/R~
=
0)
=
0
R
possesses a faithful (locally) finite dimensional polyform module
then the same is true for M,(R), the ring of n
x
n matrices
over R.
-
(2)
If
P is a torsionless left R-module such that
k
P
HomR(P,R)
=
{
2
pifi /pi
E
P,
fi
E
HomR(P,R))
=
R
@
R
i=l
-
-
faithful and cofaithful (locally) finite dimensional polyform
module then so does the ring EndRP
(3)
Tf
e
=
e2 E R
with
ReR
=
R, R has a faithful
2
does the ring eRe.
Proof.
(1) Consider the multiplication Morita context
(R,
yny
R"~), Mn(R)) where A(n) and denote modules of
row and column vectors over A, respectively. Since rR(R)
=
0
this induces a Morita context (R, R Q, Mn(R)) where Q
=
(n>
'
R
(
(R))
. This latter context satisfies the hypotheses of
the theorem.
(2)
The standard Morita context (R, P, HomR(P,R), EndRP)
satisfies the hypotheses of Lemma 5.1 and Theorem
5.3.
(3)
This is a special case of
(2).
0
We now turn our attention to polynomial rings where it will be
convenient to adopt the following notation. For M an R-module
1
we set M[x]
=
R
[~j&~
M; M[x] has a natural structure of
R[x]-
module. For any subset
N
of M[x] and integer
k
2
0,
we set
Nk={~(x)EN/degreep(x)Sk].
If
fEHom (N,M[x]) isa
R[xl

RINGS WITH FAITHFUL POLYFORM MODULES
non-zero partial endomorphism of M[x], we will set
n(f)
=
minldeg q(x)
I
0
#
q(x) ENf),
m(f)
=
minideg p(x)
I
deg (p(x))f
=
n(f)),
and L(f)
=
Nrn(,)
fl
{p(x)
I
deg(p(x))f
<
n(f)]. When the
homomorphism f is not in doubt, we will simply write n, m,
L
for n(f), m(f), L(f), respectively. Observe that
L
is an
R-submodule of Nm and flL is then a non-zero partial
R-
homomorphism from Nm into M[x],
LEMMA
5.5.
If
f
c
Hom (N, M[x]) kernel f is an essential
R[xl
R[x]-submodule of N
then
kernel flL is an essential R-submodule
Proof. Suppose that
0
#
p(x)EL.
By
hypothesis, there exists
1
r(x)
E
R [x] with
0
#
r(x)p(x)
E
kernel f. We proceed by induction
on degree r(x) to show that there then exists
0
#
a
ER'
with
This result is trivially true for degree r(x)
=
0.
So suppose
i
that degree r(x)
=
k
>
0.
Write r(x)
=
t
a.x
,
(p(x))f
=
n.
i=0
1
m.xl, where each ai
CR',
each mi€
M,
and ak
#
0.
Then
1
i=O
r(x)(p(x)f)
=
(r(x)p(x))f
=
0
so that akmn
=
0.
Hence degree
(akp(x))f
=
degree ak(p(x)f)
<
n, so by the choice of n
=
n(f),
If akp(x)
#
0
then
0
#
akp(x)
E
kernel f
lL
and we would be
k
done. So we may assume that akptx)
=
0.
Set rl (x)
=
r(x)
-
akx
.

1168
ZELMANOWITZ
0
f
rl(x)p(x)
E
kernel f with degree rl (x)
<
degree r(x)
.
The
induction hypothesis now completes the proof.
THEOREM
5.6
If
M is
a
polyform R-module then M[x]
&
polyform R[x]-module.
Proof. Let fEHom (N, M[x]) be a non-zero partial endomor-
-
R[xl
phism of M[x]. Since L
r
Nm
E
M[x]
Z
M b+l)
and
rn
Lf
r;
MIX],
Z
M as R-modules, we may regard f
lL
as a non-
zero partial endomorphism of
M(~)
where
k
=
max(m+l, n+l). By
Proposition 3.3,
M(~)
is a polyform R-module. Applying the pre-
vious lemma, we learn that kernel
f
cannot be essential in
N.
Hence M[x] is a polyform R[x]-module.
[I]
COROLLARY
5.7.
-
If R has
a
faithful locally finite dimensional
polyform module then so does the polynomial ring R[x].
Proof.
If N is a finite-dimensional R-submodule of
M
then, as
is well-known, N[x] is a finite-dimensional R[x]-submodule of
M[x]
.
Also, M[x] is clearly a faithful R[x]-module whenever M
is a faithful R-module. The result now follows from the preceding
theorem.
0
ACKNOWLEDGEMENTS
Supported in part by NSERC grant 1124543-30. The author grate-
fully acknowledges the hospitality of the McGill University
Mathematics Department, where the first draft of this manuscript
was prepared.

RINGS WITH FAITHFUL POLYFORM MODULES
REFERENCES
F.W. ANDERSON and K.R. FULLER, Rings and Categories of
Modules (Springer-Verlag, New York, 1974).
V. CAMILLO and J. ZELMANOWITZ, 'Dimension modules',
Pacific J. Math. 91 (1980), 249-261.
G. DESALE and W.K. NICHOLSON, 'Endoprimitive rings',
J. Algebra 70 (1981), 548-560.
C. FAITH, Lectures on Injective Modules and Quotient
Rings,
Lecture Notes in Math., vol. 49 (Springer-
Verlag, Berlin, 1967).
K.R. FULLER, 'Density and equivalence', J. Algebra 29
(1974), 528-550.
K. KOH, 'Quasi-simple modules and other topics in ring
theory', Lecture Notes in Math., vol. 246 (Springer-
Verlag, New York, 1972), 323-428.
J.
LAMBEK and B.A. RATTRAY, 'Localization and duality in
additive categories', Houston J. Math. 1 (1975),
87-100.
W.K. NICHOLSON. J.F. WATTERS and J.M. ZELMANOWITZ, 'On
extensions of weakly primitive rings', ~anadian J.
Math. 32 (1980), 937-944.
H.H. STORRER. 'On Goldman's primary decomposition',
Lecture.~otes in Math. ,-vol. 246 (sp;inger-verlag,
New York, 1972), 617-661.
R. WISBALER, 'Localization of modules and the central
closure of rings', Comm. in Algebra 9 (1982),
1455-1493.
J.M. ZELMANOWITZ, 'The structure of rings with faithful
nonsingular modules'. Trans. her. Math. Soc. 278
(1983), 347-359.
,
'Representations of rings with faithful
monoform modules',
J.
London Math. Soc. 29 (1984),
237-248.
Received: April 1985
Revised: November 1985