Characterizations of regular modules

Abstract

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.

Full text

International Electronic Journal of Algebra
Published Online: December 2022
DOI: 10.24330/ieja.1224782
CHARACTERIZATIONS OF REGULAR MODULES
Philly Ivan Kimuli and David Ssevviiri
Received: 14 July 2021; Revised: 28 November 2022; Accepted: 14 December 2022
Communicated by Tu˘g¸ce Pekacar C¸ alcı
Dedicated to the memory of Professor Edmund R. Puczy lowski
Abstract. Different and distinct notions of regularity for modules exist in
the literature. When these notions are restricted to commutative rings, they
all coincide with the well-known von-Neumann regularity for rings. We give
new characterizations of these distinct notions for modules in terms of both
(weakly-)morphic modules and reduced modules. Furthermore, module theo-
retic settings are established where these in general distinct notions turn out
to be indistinguishable.
Mathematics Subject Classification (2020): 16D80, 13C13, 13C05, 16E50
Keywords: Regular module, reduced module, (weakly-)morphic module
1. Introduction
Let Rbe an associative and unital ring that is not necessarily commutative and
Mbe a right R-module. We call R(unit-)regular if for each a∈Rthere exists a
(unit) y∈Rsuch that a=aya. It is strongly regular if for each a∈Rthere exists
an element y∈Rsuch that a=a2y, or equivalently if it is regular and idempotents
are central. In the literature, there are different characterizations of a regular ring
which are distinct for modules. For instance, see [28, pg. 237] and [3, Exercises 15
(13)], a ring is regular ⇔every right (left) cyclic ideal is a direct summand ⇔every
finitely generated right (left) ideal is a direct summand. Ris strongly regular ⇔
it is regular and reduced ⇔every right (left) cyclic ideal is generated by a central
idempotent ⇔it is regular and Ra ⊆aR for every a∈R⇔aR =a2Rfor each
a∈R. Where Ris commutative, it is regular ⇔it is strongly regular.
Following the (von-Neumann) regularity characterizations for rings, different au-
thors have come up with different definitions for the notion of “regularity” for mod-
ules. We outline some of them below (see also Definition 5.5, [28, Definition 2.3]
and [29]):
This work was carried out at Makerere University with support from ISP through the Makerere-
Sida Bilateral Program Phase IV, Project 316 Capacity Building in Mathematics and its Appli-
cation and through the Eastern Africa Algebra Research Group (EAALG).

2PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Definition 1.1. An R-module Mis said to be
(a) endoregular [15] and [28] if φ(M) and ker(φ) are direct summands of Mfor
every endomorphism φof M;
(b) Abelian endoregular [15] if EndR(M) is a strongly regular ring;
(c) F-regular [8] if for every submodule Nof M, the sequence 0 →NNE→
MNEis exact for each R-module E;
(d) strongly F-regular [25] if every finitely generated submodule of Mis a direct
summand of M. (In the bullets (e), (f) and (g) below, Ris commutative.)
(e) JT-regular [11] if for each m∈M, mR =M a =M a2for some a∈R;
(f) weakly JT-regular [1] if Ma =M a2for each a∈R;
(g) weakly-endoregular [2] if Ma and lM(a) are direct summands of Mfor each
a∈R.
An R-module Mis reduced [16] if whenever a∈Rand m∈Msatisfy ma2= 0,
then mRa = 0. Reduced modules are a generalisation of reduced rings. Recall that
a ring is said to be reduced [17] if it has no non-zero nilpotent elements. Thus Ris
a reduced ring if and only if Ris a reduced R-module.
We call Mamorphic module if every endomorphism φof Mhas a cokernel which
is isomorphic to its kernel, i.e., if for every endomorphism φof M, M /φ(M)∼
=
ker(φ) as R-modules. Note that the property M/φ(M)∼
=ker(φ) is the dual of
the First Isomorphism Theorem for the module endomorphism φ. This notion has
been widely studied, see for instance [21]. Recently for commutative rings, Mis
called weakly-morphic in [13] if M/Ma ∼
=lM(a) as R-modules for each a∈R, i.e.,
if every endomorphism φaof Mgiven by right multiplication by a∈Ris morphic.
It turns out that a (commutative) ring Ris right (and left) morphic if and only if
the R-module Ris a weakly-morphic module. Morphic rings have been studied in
[5,22,23].
The relationship between morphic, reduced and regular rings has been exten-
sively investigated in the literature, dating back to when Ehrlich [7] proved that a
ring is right morphic and regular if and only if it is unit-regular. Since then, the
study of the morphic property in rings has flourished due to the way morphic rings
connect with reduced rings to provide conditions related to regular rings. Recall
that a reduced ring need not be regular or (right) morphic in general. For example,
the ring of integers Zis reduced. However, it is neither (right) morphic nor regu-
lar. In general, we have the following relations about rings: strongly regular (i.e.,
regular with central idempotents) ⇔reduced and (right) morphic ⇒unit-regular
⇔(right) morphic and regular ⇒regular. Indeed by [5, Proposition 4.13], strongly
regular (i.e., regular with central idempotents) rings coincide with reduced and
(right) morphic rings, and these are unit-regular. By [7, Theorem 1], unit-regular
rings are exactly the (right) morphic and regular rings. The remaining implications

CHARACTERIZATIONS OF REGULAR MODULES 3
do not reverse in general. The ring R:= RLR(where Ris the ring of real num-
bers) is unit-regular and hence morphic. But Rhas some nontrivial idempotents
which are not central (therefore, not reduced) by [7, Corollary to Theorem 1] and
[15, Example 2.25]. The ring L∞
n=1 Ris regular but not right morphic (therefore,
not unit-regular) and the ring Z/4Zis morphic but not regular.
This paper gives new characterizations of regular modules given in Definition 1.1
in terms of (weakly-)morphic and reduced (sub-)modules. We prove that a module
is weakly-morphic and reduced if and only if it is weakly-endoregular (Theorem 2.1);
the class of Abelian endoregular modules coincides with that of morphic modules
with reduced rings of endomorphisms (Theorem 3.5); if a module Mis strongly
F-regular, then each of its submodule is invariant under every endomorphism of M
if and only if Mis a morphic module with a reduced ring of endomorphisms (The-
orem 4.6). A module is F-regular if and only if each of its (cyclic) submodules is a
weakly-morphic and reduced module (Theorem 4.12). Conditions for which one still
gets coincidence of different notions of regularity in the module theoretic setting are
established. For instance, in the subcategory of finitely generated modules, the fol-
lowing coincide: weakly-morphic and reduced ⇔F-regular ⇔weakly-endoregular
⇔weakly JT-regular (Proposition 5.8).
Notation and conventions. Throughout this paper, all rings Rwill be asso-
ciative and unital but not necessarily commutative, Mis a unitary right R-module
and Sdenotes EndR(M), the ring of endomorphisms of M. Therefore, in this case
Mcan be viewed as a left S-right R-bimodule. By Z,Qand Rwe denote the ring
of integers, rational numbers and real numbers, respectively. For φ∈S, ker(φ)
and Im(φ) denote the kernel and image of φ, respectively. The notation N⊆M
means that Nis a submodule of M. We also define rM(I) := {m∈M:I(m) =
0}, lS(I) := {φ∈S:φI = 0}, rS(I) := {φ∈S:Iφ = 0}for a nonempty subset
Iof S;rR(N) := {a∈R:Na = 0}, lS(N) := {φ∈S:φ(N) = 0}for N⊆M
and lM(A) := {m∈M:mA = 0}for A⊆R. Note that rM(φ) := ker(φ) for
0=φ∈Sand AnnR(M) := rR(M), the (right) annihilator of M. For any a∈R,
the principal ideal generated by ais denoted by (a).
The following definitions are necessary in the remaining part of this section and
will be used freely in the next sections.
Definition 1.2. A ring Ris
(a) reduced if it has no non-zero nilpotent elements;
(b) reversible if ab = 0 implies ba = 0 for any a, b ∈R;
(c) said to have Insertion-of-Factors-Property (IFP) if for a, b ∈R, ab = 0 implies
that arb = 0 for every r∈R.

4PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Definition 1.3. An R-module Mis
(a) reduced if whenever a∈Rand m∈Msatisfy ma2= 0, then mRa = 0;
(b) symmetric if whenever a, b ∈Rand m∈Msatisfy mba = 0, we have mab = 0;
(c) said to possess IFP if whenever a∈Rand m∈Msatisfy ma = 0, then
mra = 0 for each element rof R.
The notions in the Definitions 1.2 and 1.3 have been widely studied in [6,9,14,16,
17]. A module MRis said to be rigid [6] if given a∈Rand m∈M, the condition
ma2= 0 implies ma = 0. This is equivalent to lM(an) = lM(a) for every a∈Rand
n∈Z+. For a commutative ring R, it was shown in [14] that Mis reduced if and
only if lM(an) = lM(a) for every a∈Rand n∈Z+. As a dual notion to reduced
modules in [14], we have co-reduced modules.
Definition 1.4. Let Rbe a commutative ring. An R-module Mis said to be
co-reduced if Ma =Manfor every a∈Rand n∈Z+.
For noncommutative rings, we give characterizations of reduced modules and
reduced rings.
Lemma 1.5. Let Rbe a ring and Mbe a nontrivial R-module. The following
statements are equivalent:
(1) Mis reduced;
(2) Mis symmetric and lM(an) = lM(a)for every a∈Rand n∈Z+;
(3) Mhas IFP and lM(an) = lM(a)for every a∈Rand n∈Z+.
Proof. (1)⇔(2) Assume that (1) holds. By [9, Theorem 2.2], reduced modules are
symmetric. To prove that lM(an) = lM(a), let x∈lM(an). Then xan= 0. As M
is reduced, xRa = 0 and so xa = 0. This gives lM(an)⊆lM(a). Since the reverse
inclusion is trivial, we obtain lM(an) = lM(a). The proof of (2) ⇒(1) holds after
applying [9, Corollary 2.2].
(3)⇔(1) This follows from [6, Proposition 2.8]. □
Corollary 1.6. The following statements are equivalent for a ring R:
(1) Ris reduced,
(2) lR(an) = lR(a)for every a∈Rand n∈Z+,
(3) rR(an) = rR(a)for every a∈Rand n∈Z+.
Proof. (1)⇔(2) Reduced rings are reversible and hence have IFP. By Lemma 1.5,
(2) follows from (1). Conversely, let an= 0. Then 1R∈lR(an) = lR(a), so
a= 1R·a= 0. This proves that Ris reduced. The proof of (1)⇔(3) is similar. □
If MRis a reduced module over a commutative ring R, then M a ∼
=Manfor
each a∈Rand n∈Z+. To see this, assume that Mis reduced. By Lemma 1.5,

CHARACTERIZATIONS OF REGULAR MODULES 5
lM(an) = lM(a), and so Ma ∼
=M/lM(a) = M/lM(an)∼
=Man. For a not necessar-
ily commutative ring R, the map φ:M→Mgiven by m7→ ma for a∈Rneed
not be an endomorphism. We show in Proposition 1.7 that when Mis reduced and
eis an idempotent element of R, then m7→ me is an idempotent endomorphism of
M.
Proposition 1.7. Let Rbe a ring, Mbe a reduced R-module, m∈Mand e2=
e∈R. Then every map φedefined by φe(m) = me is an idempotent element of S.
Proof. Let ebe an idempotent element in Rand m∈M. Since Mis reduced, it
has IFP and so me(1R−e) = 0 implies that for any r∈R, mer(1R−e) = 0, that is,
mer =mere. On the other hand, m(1R−e)e= 0 implies that mre =mere. Hence
mer =mre. Now for all r∈R, φe(mr)=(mr)e= (me)r=φe(m)r. Closure under
addition always holds. □
2. Weakly-endoregular modules
Lee, Rizvi & Roman [15] and Ware [28, Corollary 3.2] call Mendoregular if
EndR(M) is a regular ring. To study the various regularity properties of the rings
of endomorphisms, Anderson and Juett [2] defined weakly-endoregular modules. A
module Mover a commutative ring Ris weakly-endoregular if and only if for each
a∈R, M =Ma LlM(a). We give a characterization of weakly-endoregular mod-
ules in terms of weakly-morphic and (co-)reduced modules. For other equivalent
statements of Theorem 2.1 see [2, Theorem 1.1].
Theorem 2.1. Let Rbe a commutative ring and Mbe a nontrivial R-module.
The following statements are equivalent:
(1) Mis weakly-morphic and reduced,
(2) Mis weakly-morphic and co-reduced,
(3) Mis co-reduced and reduced,
(4) Mis weakly-endoregular.
Proof. (1)⇒(2) Assume that (1) holds. We need to show that Ma =Manfor
every a∈Rand n∈Z+. Since Mis weakly-morphic, M/M a ∼
=lM(a) and
M/Man∼
=lM(an). By Lemma 1.5, lM(a) = lM(an), and so M/M a ∼
=M/Man.
Therefore there exists an isomorphism φsuch that φ(M/Man) = M/Ma. This,
φ(Ma/Man) = φ(M/M an)a=M a/Ma = 0. So M an=Ma, as desired.
(2)⇒(1) Assume that (2) holds. Then Ma =Manfor every a∈Rand n∈Z+.
Since Mis weakly-morphic, lM(a)∼
=M/Ma =M/Man∼
=lM(an). This gives
lM(a)∼
=lM(an). In view of [14, Proposition 2.3], it remains to prove that this is
equality. By [13, Lemma 1], there exists some φ∈Ssuch that lM(an) = φ(M)
and ker(φ) = Man. The two equalities imply that 0 = φ(M)an=φ(Man). By

6PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
hypothesis, Ma =Man. So φ(Man) = φ(Ma) = φ(M)a, from which we deduce
that lM(an) = φ(M)⊆lM(a).Hence lM(an) = lM(a).
(2)⇒(3) Follows from the proof of (2) ⇒(1).
(3)⇒(4) Let a∈Rand assume (3). Then M a =M a2by Definition 1.4 and
lM(a) = lM(a2) by Lemma 1.5. It follows from [2, Theorem 1.1] that Mis weakly-
endoregular.
(4)⇒(1) Since M=M a LlM(a), M/Ma ∼
=lM(a) for every a∈R. Thus M
is weakly-morphic. Next, let x∈M=M a LlM(a) such that xa2= 0. Then
(xa)a= 0 and so xa ∈lM(a). But also xa ∈M a and Ma ∩lM(a) = 0. Therefore,
xa = 0 and Mis a reduced module. □
Corollary 2.2. Let Rbe a commutative ring and Mbe a nontrivial finitely gen-
erated R-module. Then the following statements are equivalent:
(1) Mis weakly-morphic and reduced,
(2) Mis co-reduced,
(3) R/AnnR(M)is a regular ring,
(4) Mis weakly-endoregular.
Proof. (1)⇒(2) Follows from Theorem 2.1.
(2)⇒(3) Suppose that M a =M anfor each a∈Rand n∈Z+. Then M(a) =
M(a)(an) with M(a) a finitely generated module. Using [4, Corollary 2.5], we have
M(a)(1 + (an)) = 0, which implies that M(a)(1 + anr) = 0 for all r∈R. It
then follows that (a+an+1r)∈AnnR(M) and hence a+ AnnR(M) = an+1(−r) +
AnnR(M)∈(an+1) + AnnR(M). This gives (a) + AnnR(M)⊆(an+1) + AnnR(M)
and, consequently, (a) + AnnR(M) = (an+1) + AnnR(M). Since a=ra2+s
for some r∈Rand s∈AnnR(M), a −ra2∈AnnR(M) and a=ra2for some
r∈R:= R/AnnR(M), we have R/AnnR(M) regular.
(3)⇒(4) Assume that (3) holds. Using this assumption and the First Isomor-
phism Theorem for the R-endomorphism φ:R→S:= EndR(M) defined by
φ(a) = φafor all a∈R, we obtain {φa:a∈R}is regular. By [13, Proposition 7],
Mis a weakly-endoregular module.
(4)⇒(1) Follows from Theorem 2.1. □
Corollary 2.3. The following are equivalent for a commutative ring R:
(1) Ris morphic and reduced,
(2) Ris co-reduced,
(3) Ris regular,
(4) R= (a)LrR(a)=(a)LlR(a)for each a∈R.

CHARACTERIZATIONS OF REGULAR MODULES 7
Proof. (2)⇔(3) A commutative ring Ris regular if and only if for each a∈
R, aR =a2R. Thus Ris co-reduced if and only if it is regular. The equivalences
(1)⇔(3)⇔(4) follow from Theorem 2.1. □
Corollary 2.4. Every module over a commutative regular ring is weakly-endoregular.
Proof. This follows from [13, Proposition 13]. □
3. Abelian endoregular modules
The focus of this section is the characterization of Abelian endoregular modules
in terms of reduced and morphic modules. A ring Ris said to be Abelian if all
its idempotents are central. If Ris reduced, then every idempotent is central. A
strongly regular ring is reduced, regular and Abelian. More generally, an R-module
Mis said to be Abelian if Sis an Abelian ring. MRis an Abelian endoregular
module if Sis a regular and Abelian ring.
Remark 3.1. MRis an Abelian endoregular module if and only if M=φ(M)Lker(φ)
for every φ∈S. Abelian endoregular modules are morphic modules. Note that an
endoregular module need not be morphic. The Z-module L∞
i=1 Qi, where Qi=Q,
is endoregular but not morphic.
Recall that Mcogenerates M/φ(M), φ ∈Sif M/φ(M) can be embedded in
M(I), where Iis an index set. That is, 0 =x∈M/φ(M), φ ∈S, implies that
γ(x)= 0 for some γ∈HomR(M/φ(M), M ) [20, pg. 230].
Lemma 3.2. If Ris a ring and Mis a nontrivial morphic R-module, then for
every φ∈S,
φ(M) = rM(lS(φ)).
Moreover, the following statements are equivalent:
(1) φ(M) = rM(lS(φ)) for every φ∈S;
(2) For each φ∈Sand m∈M, if lS(φ(M)) ⊆lS(m), then m∈φ(M);
(3) Mcogenerates M/φ(M)for each φ∈S.
Proof. For every φ∈S, there exists ψ∈Ssuch that φ(M) = ker(ψ) = rM(ψ)
and ψ(M) = ker(φ) = rM(φ). It follows from the equality φ(M) = rM(ψ) that
ψφ(M) = 0 which gives ψφ = 0 and hence S ψ ⊆lS(φ(M)). Thus rM(lS(φ)) ⊆
rM(ψ) = φ(M). The reverse inclusion is obvious, hence rM(lS(φ)) = φ(M).
(1)⇒(2) Let m∈Mand φ∈Ssuch that lS(φ(M)) ⊆lS(m). Then m∈
rM(lS(m)) ⊆rM(lS(φ(M))) = φ(M) by (1). Hence m∈φ(M).
(2)⇒(1) Clearly φ(M)⊆rM(lS(φ(M))) for every φ∈S. Let m∈rM(lS(φ(M))).
Then we have lS(rM(lS(φ(M)))) ⊆lS(m). By [3, Proposition 24.3], lS(φ(M)) ⊆
lS(m). By (2), m∈φ(M).

8PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
(1)⇔(3) Assume that (1) holds and let φ∈S. In view of [3, pg. 109 and Lemma
24.4],
RejM/φ(M)(M) := \{ker(γ) : γ∈HomR(M/φ(M), M )}
=rM(lS(φ(M)))/φ(M)
= 0
for each φ∈S. Applying [3, Corollary 8.13], Mcogenerates M/φ(M) for each
φ∈S. Conversely, suppose Mcogenerates M/φ(M) for each φ∈S. Then
RejM/φ(M)(M) = 0 for each φ∈Sby [3, Corollary 8.13]. Applying [3, Lemma
24.4] gives rM(lS(φ(M)))/φ(M) = 0. Hence rM(lS(φ(M))) = φ(M) follows. □
Remark 3.3.
(a) In view of Lemma 3.2, the hypothesis “φ(M) = rM(lS(φ(M)))” in statement
(b) of [15, Proposition 4.2] is superfluous.
(b) Recall that in [20], a module SMis P-injective if φ(M) = rM(lS(φ)) for every
φ∈S. A ring Ris called left P-injective if it is a P-injective right R-module
(equivalently, rRlR(a) = aR for every a∈R). Thus, if Mis a morphic R-
module, then SMis a P-injective module by Lemma 3.2.
Lemma 3.4. If Mis a morphic R-module and Sis a reduced ring, then for every
φ∈S,
rM(φ2) = rM(φ).
Proof. We only prove rM(φ2)⊆rM(φ) since the reverse inclusion is obvious. Since
Mis morphic, there exists γ∈Ssuch that γ(M) = rM(φ2). This implies φ2γ= 0.
Further, Sbeing reduced implies that φγ = 0. So, γ(M)⊆rM(φ) and we get
rM(φ2) = γ(M)⊆rM(φ). □
Now we give a characterization of Abelian endoregular modules in terms of mor-
phic modules and reduced rings of endomorphisms.
Theorem 3.5. Let Rbe a ring and Mbe a nontrivial R-module. The following
statements are equivalent:
(1) MRis a morphic module and Sis a reduced ring,
(2) MRis an Abelian endoregular module.
Proof. (1)⇒(2) Since Sis reduced, lS(φ) = lS(φ2) for any φ∈Sby Corollary 1.6.
Applying Lemma 3.2, φ(M) = rM(lS(φ(M))) = rM(lS(φ2(M))) = φ2(M). This
gives φ(M) = φ2(M). For any m∈M, φ(m)∈φ(M) = φ2(M), and so φ(m) =
φ(n) for some n∈φ(M). Therefore, x:= m−n∈rM(φ) and m=x+n∈
rM(φ) + φ(M). We obtain M=rM(φ) + φ(M). To prove that this is a direct
sum, let y∈rM(φ)∩φ(M). Then y=φ(m) for some m∈Mwith φ(y) =

CHARACTERIZATIONS OF REGULAR MODULES 9
φ2(m) = 0. Consequently we have m∈rM(φ2) = rM(φ) by Lemma 3.4, from
which we have y=φ(m) = 0, thus we obtain 0 = rM(φ)∩φ(M). This proves
M=rM(φ)Lφ(M), and Mis an Abelian endoregular module.
(2)⇒(1) Mis morphic by Remark 3.1. In addition, since Sis a strongly regular
ring, it is reduced. □
Corollary 3.6. Let Rbe a ring and Ma nontrivial R-module. The following
statements are equivalent:
(1) MRis morphic and SMis reduced,
(2) MRis an Abelian endoregular module.
Proof. (1)⇒(2) Let φ∈Ssuch that φ2= 0. Then φS(m) = 0 for all m∈Mby
(1). It follows that φ1M(m) = φ(m) = 0 for all m∈M, so φ= 0. This shows that
Sis a reduced ring and (2) follows by Theorem 3.5.
(2)⇒(1) MRis clearly morphic by Remark 3.1. Let φ∈Sand m∈Msuch
φ2(m) = 0. Then φ(φ(m)) = 0. In view of Remark 3.1, φ(m)∈rM(φ)∩φ(M) = 0,
so φ(m) = 0. Since Sis strongly regular, there exists some ψ∈Ssuch that
φ=φψφ with φψ =ψφ a central idempotent element of S. Thus φS(m) =
φψφS(m) = φSψφ(m) = 0. This proves that SMis a reduced module. □
By considering the case M=Rand EndR(R)∼
=R, we have:
Corollary 3.7. The following statements are equivalent for a ring R:
(1) Ris right morphic and reduced,
(2) Ris left P-injective and reduced,
(3) R=aR LrR(a)for each a∈R,
(4) Ris strongly regular.
Proof. (1)⇒(2) This follows by Lemma 3.2 and Remark 3.3 (b).
(1)⇔(3) and (3)⇔(4) These are a consequence of Theorem 3.5 and Remark 3.1.
(2)⇒(4) Since Ris reduced, for each a∈R, lR(a) = lR(a2) by Corollary 1.6.
It follows that aR =rR(lR(a)) = rR(lR(a2)) = a2Rbecause Ris left P-injective.
Thus a=a2yfor some y∈R, and this proves Ris strongly regular. □
A module MRis duo provided every submodule of Mis fully invariant, that is,
for any submodule Nof M, φ(N)⊆Nfor every φ∈S. A ring Ris right duo if
every right ideal of Ris a two-sided ideal, equivalently if Ra is contained in aR for
every element ain R[24].
Lemma 3.8. [24, Lemma 1.1] Let Rbe any ring. Then a right R-module Mis a
duo module if and only if for each endomorphism φof Mand each element mof
Mthere exists ain Rsuch that φ(m) = ma.

10 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Lemma 3.9. Let Rbe a commutative ring. For a nontrivial duo R-module M,
consider the following statements:
(1) Mis reduced as a right R-module,
(2) Mis reduced as a left S-module,
(3) Sis a reduced ring.
Then (1) ⇔(2) ⇒(3).
Proof. (1)⇒(2) Let φ∈Sand m∈Msuch that φ2(m) = 0. Then ma2= 0 for
some a∈Rbecause Mis duo. By (1), mra = 0 for all r∈R. Since every element
in Sis defined by right multiplication of each element of Mby some element of
R, φ(ψ(m)) = mra = 0 for every ψ∈Sfor some r∈R. Thus φS(m) = 0 and SM
is a reduced module.
(2)⇒(1) Let m∈Mand a∈Rsuch that ma2= 0. Then the endomorphism
φ:M→M, x 7→ xa gives φ2(m) = 0. Since SMis reduced, we have φS(m) = 0.
Note that for every r∈R, right multiplication by rdefines an endomorphism ψr:
M→M, m 7→ mr. This gives mra =φψr(m) = 0. Since mRa ⊆φS(m) = 0, MR
is a reduced module.
(2)⇒(3) Let φ∈Ssuch that φ2= 0. Then for each m∈M, Lemma 3.8 gives
0 = ma2=φ2(m) for some a∈R; and so φS(m) = 0 by (2). It follows that
φ1M(m) = φ(m) = 0. Since mwas chosen arbitrarily, φ= 0. □
Note that even when a duo module has a reduced ring of endomorphisms, the
module itself may not be reduced.
Example 3.10. Let R:= Zbe a ring. For any prime p, the Pr¨ufer p-group
M:= Z(p∞) is an Artinian uniserial R-module and hence a duo module by [24,
pg. 536]. Then it is well-known that S:= EndR(M) is the ring of p-adic integers
[3, Exercises 3 (17), pg. 54]. Since the ring of p-adic integers is a commutative
domain, it is a reduced ring. However, Mis neither reduced as an R-module nor
as an S-module.
An R-module Mis said to be a multiplication module provided for each submod-
ule Nof Mthere exists an ideal Aof Rsuch that N=MA. Finitely generated
multiplication modules are duo.
Lemma 3.11. [13, Proposition 19] Let Mbe a finitely generated multiplication
module over a commutative ring R. Then Mis weakly-morphic if and only if it is
morphic.
Corollary 3.12. Every cyclic module over a commutative ring Ris weakly-
morphic if and only if it is morphic.
Proof. Since every cyclic R-module is a multiplication module that is finitely gen-
erated, it is weakly-morphic if and only if it is morphic by Lemma 3.11. □

CHARACTERIZATIONS OF REGULAR MODULES 11
Proposition 3.13. Let Rbe a commutative ring and Mbe a nontrivial finitely
generated multiplication R-module. Then Mis weakly-endoregular if and only if it
is Abelian endoregular.
Proof. Assume that MRis a weakly-endoregular module. By Theorem 2.1, MRis
weakly-morphic and reduced. As it is a finitely generated multiplication module,
Lemma 3.11 implies MRis morphic. Being duo, SMis a reduced module by
Lemma 3.9. Applying Corollary 3.6 proves that Mis Abelian endoregular. The
converse clearly holds since every Abelian endoregular module over a commutative
ring is weakly-endoregular. □
Corollary 3.14. Every cyclic module over a commutative ring Ris weakly-
endoregular if and only if it is Abelian endoregular.
Proof. Since cyclic modules are finitely generated multiplication modules, the
proof of the corollary is immediate from Proposition 3.13. □
An R-module Mis strongly duo [12] if the trace of Min Nis N, that is,
TrN(M) := P{Im(λ) : λ∈HomR(M , N)}=Nfor all N⊆MR. Clearly, ev-
ery strongly duo module Mis a duo module. In [12, Theorem 5.5], the ring of
endomorphisms of a module Mthat is strongly duo and reduced was shown to
be a strongly regular ring. For commutative rings, we have an improved result in
Corollary 3.15.
Corollary 3.15. Let Rbe a commutative ring and Mbe a nontrivial duo R-
module. The following statements are equivalent:
(1) MRis a morphic and reduced module,
(2) Sis a strongly regular ring.
Proof. (1)⇒(2) Assume (1) holds. Then Sis a reduced ring by Lemma 3.9 and
is, therefore, strongly regular by Theorem 3.5.
(2)⇒(1) MRis morphic by Remark 3.1 and reduced by Lemma 3.9. □
4. F-regular modules
Recall that a ring Ris regular if and only if every right (left) cyclic ideal of
Ris a direct summand of RR. To generalize this characterization to modules,
Ramamurthi and Rangaswamy in [25, pg. 246] defined strongly regular modules.
A module Mis called strongly regular (in the sense of [25]) if every finitely generated
submodule is a direct summand, or equivalently every cyclic submodule is a direct
summand. Following Naoum [19], we call the strongly regular modules strongly F-
regular (even without commutativity of R). In [21], a relationship between morphic
finitely generated strongly F-regular modules and their rings of endomorphisms was
established.

12 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Proposition 4.1. [21, Corollary 2.7] A finitely generated strongly F-regular module
Mis morphic if and only if Sis morphic and regular.
An R-module Mis said to be k-local-retractable (for kernel-local-retractability)
(or equivalently, P-flat over S:= EndR(M)) if for any φ∈Sand any nonzero
element x∈rM(φ), there exists a homomorphism ψx:M→rM(φ) such that x∈
ψx(M)⊆rM(φ) ([15, pg. 4069] and [20]). The module MRis called a self-generator
in [20, pg. 228] if it generates each of its images, that is, mR = HomR(M, mR)(M)
for all m∈M. In this case, for each m∈M, m =Pαi(xi) with xi∈Mand
αi∈HomR(M, mR).
Proposition 4.2. Every nontrivial strongly F-regular module MRis a k-local-
retractable module.
Proof. Since strongly F-regular modules are self-generator modules by [20, pg.
228], MRis P-flat over Sby [20, Lemma 1], which is equivalent to being k-local-
retractable by [15, pg. 4069]. □
Lemma 4.3. If Mis a k-local-retractable R-module and Sis a reduced ring, then
for every φ∈S,
rM(φ2) = rM(φ).
Proof. Let x∈rM(φ2). Due to k-local-retractability of M, there exists 0 =ψx∈S
such that x∈ψx(M)⊆rM(φ2). Hence φ2ψx= 0. Since Sbeing reduced implies
φψx= 0, x ∈ψx(M)⊆rM(φ). This shows that rM(φ2)⊆rM(φ). The reverse
inclusion is well-known. □
Lemma 4.4. If Mis a nontrivial duo and strongly F-regular R-module, then S
is a reduced ring.
Proof. Let φ∈Ssuch that φ2= 0. If φ= 0, then there exists some 0 =m∈M
such φ(m)= 0. By the strongly F-regular hypothesis, M=φ(m)RLXfor some
submodule Xof M. Since Mis duo, φ(M) = φ(φ(m)R)Lφ(X) = φ(X)⊆X,
so φ(M)⊆X. This implies that φ(m)∈φ(m)R∩X= 0, a contradiction. Thus
φ= 0 and Sis a reduced ring. □
Lemma 4.5. Let Mbe a nontrivial duo and strongly F-regular R-module. If
K∼
=K′where Kand K′are submodules of M, then K=K′.
Proof. First, we prove that for every submodule Nof M, φ(N)⊆Nfor all homo-
morphisms φ:N→M. Let n∈Nand consider φ:nR →M. By the strongly
F-regular hypothesis, M=nR LXfor some submodule X. Define β:M→M
by β(s+x) = φ(s) for every s∈nR and x∈X. Then βis a well-defined endo-
morphism of Mwhich extends φto an endomorphism of M. It follows that for

CHARACTERIZATIONS OF REGULAR MODULES 13
any n∈Nthere exists β∈Ssuch that φ(n)∈φ(nR) = β(nR)⊆Nbecause Mis
duo. Hence φ(N)⊆N. Therefore, if σ:K→K′is the given isomorphism, then
K′=σ(K)⊆Kand K=σ−1(K′)⊆K′. This proves that K=K′.□
The following equivalent conditions were established in [5, Proposition 4.13 and
Lemma 4.2] for near-rings, so they must hold for rings: reduced and right morphic
⇔regular and right duo ⇔reduced and regular ⇔strongly regular. In the next
theorem we write down these ideas in the module-theoretic context.
Theorem 4.6. Let Rbe a ring and Mbe a nontrivial strongly F-regular module.
Then the following statements are equivalent:
(1) MRis a morphic module and Sis a reduced ring,
(2) MRis a duo module,
(3) MRis an Abelian endoregular module.
Proof. (1)⇒(2) Assume that (1) holds. Let Nbe a submodule of Mand φ∈S. By
the strongly F-regular hypothesis, for every n∈N, nR =e(M) for some idempotent
e∈S. Since Sis reduced, eis central in S. Hence, φ(n)∈φ(nR) = φ(e(M)) =
e(φ(M)) ⊆e(M) = nR ⊆N. This proves that φ(N)⊆Nfor all φ∈S, so MRis
duo.
(2)⇒(1) Assume that (2) holds. Then Sis a reduced ring by Lemma 4.4. To
prove Mis morphic, in view of Theorem 3.5, we will show that M=φ(M)LrM(φ)
for each φ∈S. Let φ∈S. Using Lemma 4.3 and the First Isomorphism Theorem,
φ(M)∼
=M/rM(φ) = M/rM(φ2)∼
=φ2(M). This gives φ(M)∼
=φ2(M). Applying
Lemma 4.5 gives φ(M) = φ2(M). For any x∈M, φ(x)∈φ(M) = φ2(M) which
implies that there exists y∈Msuch that φ(x) = φ2(y). Then φ(x−φ(y)) = 0.
This implies that k:= x−φ(y)∈rM(φ), hence x=φ(y) + k∈φ(M) + rM(φ)
and M=φ(M) + rM(φ). Let x∈rM(φ)∩φ(M). Then x=φ(m) for some
m∈Mwith φ(x) = φ2(m) = 0. Consequently, in view of Proposition 4.2 and
Lemma 4.3, we have m∈rM(φ2) = rM(φ), from which we have x=φ(m) = 0.
Thus 0 = rM(φ)∩φ(M) and M=φ(M)LrM(φ).
(1)⇔(3) Follows from Theorem 3.5. □
Definition 4.7. A submodule Nof Mis pure in Mif the sequence 0 →NNE→
MNEis exact for each R-module E.Nis relatively divisible-pure or RD-pure in
Min case Na =Ma ∩N(equivalently, 0 →NNR/aR →MNR/aR is exact)
for each a∈R.
By [8, Proposition 8.1], every pure submodule is also RD-pure. A ring Ris
regular if and only if every (right) ideal is pure (see [8]). Using this fact, Fieldhouse
calls MRaregular module if every submodule Nof Mis pure. Following Naoum
[19], we call the Fieldhouse regular modules F-regular.

14 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Lemma 4.8. Let Rbe a commutative ring and Mbe a nontrivial F-regular module.
Then Mis weakly-endoregular, weakly-morphic, reduced and co-reduced.
Proof. Since submodules of F-regular modules are RD-pure by [8, Proposition
8.1], we show that M=Ma LlM(a) for each a∈R. Let a∈R. Then Ma =
Ma ∩Ma =Ma2so that Ma =Ma2.It follows that for any x∈M, xa =na2for
some n∈M. Since (x−na)a= 0, x−na ∈lM(a) and x=na+x−na ∈M a+lM(a).
Hence M=Ma +lM(a). By the RD-pure property, 0 = lM(a)a=M a ∩lM(a) for
every a∈R. Thus M=Ma LlM(a). This proves that Mis weakly-endoregular.
Using Theorem 2.1, Mis weakly-morphic, reduced and co-reduced. □
Example 4.9. The converse of Lemma 4.8 does not hold in general. The Z-
module Qis weakly-endoregular, weakly-morphic and reduced but it is not F-
regular. In particular, not all its submodules are (RD-)pure since 2Q∩Z= 2Zfor
the submodule Z.
Since submodules of strongly F-regular modules are RD-pure by [25, pg. 240
and 246], it follows from [13, Proposition 8] that if Ris a commutative ring, then
every strongly F-regular module is a weakly-morphic module. An R-module M
is finitely presented (abbreviated as f.p.) if there exists an exact sequence of the
form Rn→Rm→Mwith n, m ∈Z+, or equivalently if M∼
=P/Q, where Pand
Qare finitely generated modules, and Pis a projective module. Clearly, strongly
F-regular modules are F-regular but the converse is not true in general, see [1]. In
Proposition 4.11, we determine when the F-regular modules are strongly F-regular.
Lemma 4.10. [18, Theorem 7.14] If Nis a pure submodule of Mand M/N is
finitely presented, then Nis a direct summand of M.
Proposition 4.11. Let Rbe a commutative ring and Mbe a nontrivial R-module.
Then Mis strongly F-regular whenever Mis F-regular and M/mR is finitely pre-
sented for each m∈M.
Proof. Suppose Mis F-regular and M/mR is finitely presented for each m∈M.
Then mR is a pure submodule in Mfor each m∈M. By Lemma 4.10, mR is a
direct summand of Mfor each m∈Mand, thus Mis strongly F-regular. □
Let Rbe a commutative ring and Nbe a proper submodule of MR.Nis a
prime submodule if for any a∈Rand m∈M, ma ∈Nimplies either m∈N
or a∈(N:RM) := {r∈R:Mr ⊆N}. For any proper submodule Nof
M, the intersection of all prime submodules of Mcontaining Nis denoted by
Rad(N). Theorem 4.12 gives some new characterizations for F-regular modules
over commutative rings. For other equivalent statements of Theorem 4.12 see [1,

CHARACTERIZATIONS OF REGULAR MODULES 15
Theorem 6], [10, Theorem 2.3, Corollary 2.7 and Theorem 4.1] and [26, Theorem
2.1].
Theorem 4.12. Let Rbe a commutative ring and Mbe a nontrivial R-module.
The following statements are equivalent:
(1) Mis F-regular,
(2) Every submodule of Mis a weakly-endoregular module,
(3) Every submodule of Mis a weakly-morphic and reduced module,
(4) Every cyclic submodule of Mis a (weakly-)morphic and reduced module,
(5) Every cyclic submodule of Mis a co-reduced module,
(6) Every cyclic submodule of Mis an Abelian endoregular module,
(7) Every cyclic submodule of Mis an F-regular module.
Proof. (1)⇒(2) Assume that (1) holds and let Nbe a submodule of M. By
Lemma 4.8, Mis weakly-endoregular. Since Nis pure in M, by [2, Theorem 1.1
(3)] Nis weakly-endoregular as well.
(2)⇒(3) This follows from Theorem 2.1.
(3)⇒(4) Using Corollary 3.12, weakly-morphic cyclic modules are morphic.
(4)⇒(5) Since every cyclic submodule of Mis a finitely generated R-module,
the proof follows from Corollary 2.2.
(5)⇒(1) Suppose that (5) holds. Let Nbe a proper submodule of M. In view
of [10, Theorem 2.3], we have to prove that Rad(N) = N. But to prove that
Rad(N) = N, by [26, Theorem 2.1], it is enough to show that m(a) = m(a2) for
all a∈Rand m∈M. Let a∈Rand m∈M. Since mR is a co-reduced module,
mRa =mRa2. Thus m(a) = m(a2).
(2)⇒(6) Assume (2) holds. Since mR is a finitely generated multiplication mod-
ule where m∈M, it is weakly-endoregular if and only if it is Abelian endoregular
by Proposition 3.13.
(6)⇒(5) Assume that mR is Abelian endoregular module for each m∈M. Then
mR =mRa LlmR(a) for each m∈Mand a∈R. It follows that mRa =mRa2,
and mR is co-reduced for each m∈Mby Definition 1.4.
(1)⇒(7) Assume that Mis F-regular. Then mR is an F-regular module for each
m∈Mby [8, Theorem 8.2] and [10, Proposition 2.6].
(7)⇒(1) Assume mR is an F-regular module for each m∈M. Then by Defi-
nitions 1.1 and 4.7, mRa is a(n) (RD-)pure submodule of mR for each a∈R. It
follows that mRa =mRa ∩mRa =mRa2, proving that m(a) = m(a2). By [26,
Theorem2.1], Rad(N) = Nfor each submodule Nof M. Hence Mis F-regular by
[10, Theorem 2.3]. □

16 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Remark 4.13.
(a) If Mis an F-regular module over a commutative ring, then by Lemma 4.8 and
Theorem 4.12, ker(φ) and Im(φ) are weakly-morphic and reduced modules for
every φ∈S.
(b) By Theorem 4.12, the properties: “weakly-morphic module” and “reduced
module” transfer from a module to each of its submodules and conversely.
(c) It is shown in Theorem 4.12 that if every (cyclic) submodule of Mis (weakly-)
morphic and reduced module, then Mattains the F-regularity property.
Recall that a commutative ring Ris regular if and only if for each a∈R, aR =
a2R. For commutative rings R, Jayaram and Tekir in [11] call MRregular if for
each m∈M, mR =Ma =M a2for some a∈R. Following [1, Definition 1], we
call the Jayaram and Tekir regular modules JT-regular. Anderson, Chun & Juett
in [1] defined a weak version of these modules, the weakly JT-regular modules. M
is a weakly JT-regular module if M a =M a2for each a∈R.
Remark 4.14. Let Rbe a commutative ring and Mbe an R-module.
(a) By [1, Theorem 13], Mis JT-regular ⇒Mis strongly F-regular ⇒Mis F-
regular ⇒Mis weakly JT-regular.
(b) By Definitions 1.1 and 1.4, the weakly JT-regular modules and the co-reduced
modules are indistinguishable. Therefore, by (a) F-regular (resp., strongly F-
regular, JT-regular) modules are co-reduced modules.
(c) The fact that finitely generated JT-regular modules are reduced was proved
in [11, Lemma 10]. By Lemma 4.8, if Mis an F-regular (resp., strongly F-
regular, JT-regular) module, then it is weakly-endoregular, weakly-morphic
and reduced. Since all the other forms are F-regular by (a), they are weakly-
morphic, reduced and co-reduced as well.
Corollary 4.15. Let Rbe a commutative ring and Mbe a nontrivial R-module.
Then Mis strongly F-regular whenever for each m∈M , M/mR is finitely presented
and any one of the following statements is satisfied:
(1) mR is (weakly-)morphic and reduced,
(2) mR is Abelian endoregular,
(3) mR is weakly-endoregular,
(4) mR is weakly JT-regular,
(5) mR is co-reduced,
(6) mR is F-regular.
Proof. In view of Proposition 4.11 and the fact in Remark 4.14 (b) that the weakly
JT-regular modules are the co-reduced modules, it is enough to prove that each one
of the given statements (1) to (6) implies Mis F-regular. Assume that for each

CHARACTERIZATIONS OF REGULAR MODULES 17
m∈M, mR satisfies any one of the statements given. Then Mis F-regular by
Theorem 4.12. □
Let Rbe a commutative ring and Mbe a nontrivial R-module. Table 1 illus-
trates how the properties: “(weakly-)morphic module” and “(co-)reduced module”
transfer from a module to each of its cyclic submodules and conversely. Further,
the table shows how these properties determine the nature of regularity possessed
by a module.
Table 1. Regular, (weakly-)morphic and reduced (cyclic) submodules
Mis ⇒Mis ⇒Mis ⇒Mis
Strongly F-regular weakly-endoregular weakly
F-regular JT-regular
⇑ ⇕ ⇕ ||
∀m∈M, ⇒ ∀m∈M, ⇒Mis ⇒Mis
mR is weakly-morphic mR is weakly-morphic weakly-morphic co-reduced
+mR is reduced + mR is reduced + reduced
+M/mR is f.p
⇕ ⇕ ⇕ ||
∀m∈M, ⇒ ∀m∈M, ⇒Mis ⇒Mis
mR is co-reduced mR is co-reduced weakly-morphic co-reduced
+M/mR is f.p + co-reduced
Example 4.16. The implications in the rows of Table 1 cannot be reversed in
general.
(a) Co-reduced ⇒ weakly-morphic. Let pbe a prime element of Z. Then the
Pr¨ufer p-group Zp∞is a co-reduced Z-module. However, since any non-zero
endomorphism of the type φaof Zp∞is surjective but not injective, Zp∞is not
weakly-morphic as a Z-module (see [13, Proposition 4 and Example 2.2]).
(b) Weakly-morphic + (co-)reduced on M⇒ weakly-morphic on cyclic submodules
module of M. The Z-module Qis weakly-morphic, (co-)reduced. However, its
cyclic Z-submodule Zis not weakly-morphic.
(c) Co-reduced (= Weakly-morphic + reduced) on mR, m ∈M⇒ M/mR is
finitely presented. Teply in [27] constructed a commutative regular ring Rwith
a finitely generated F-regular R-module Mhaving a submodule T(M) := {m∈
M:rR(m) is an essential ideal of R}.By Theorem 4.12, each mR, m ∈Mis
weakly-morphic, reduced and co-reduced. However, since by [27] T(M) is a

18 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
cyclic pure submodule which is not a direct summand of M,M/T (M) is not
finitely presented by Lemma 4.10.
Corollary 4.17. Let Rbe a commutative ring and Man R-module. Mis weakly-
endoregular if and only if it is weakly JT-regular and weakly-morphic if and only if
it is weakly-morphic and reduced.
Proof. Since weakly JT-regular modules are exactly the co-reduced modules, the
proof follows by Theorem 2.1. □
5. Coincidence of morphic, reduced and regular modules
This section gives conditions under which the different regularity notions of mod-
ules coincide with weakly-morphic and reduced modules. Further, under some spe-
cial conditions, we give the kind of regularity a module will attain whenever every
(cyclic) submodule of such a module is (weakly-)morphic and reduced. Note that
(using Lemma 4.8, Example 4.9, Remark 4.14 and [1, pg. 15 & Example 35 (6)])
F-regular ⇒weakly-endoregular ⇒ F-regular ⇒weakly JT-regular ⇒ F-regular.
Theorem 5.1. Let Rbe a commutative ring and Mbe a nontrivial finitely gen-
erated R-module. Then the following statements are equivalent:
(1) Mis weakly-morphic and reduced,
(2) R/AnnR(M)is a regular ring,
(3) Mis weakly-endoregular,
(4) Mis weakly JT-regular,
(5) Mis F-regular,
(6) Every cyclic submodule of Mis a (weakly-)morphic and reduced (resp.,
weakly-endoregular, Abelian endoregular, co-reduced, weakly JT-regular, F-
regular) module.
Proof. (1)⇔(3) Follows from Theorem 2.1.
(1)⇔(2) ⇔(4) Follows Corollary 2.2 and Remark 4.14 (b), respectively.
(3)⇔(5) Follows from [1, Theorem 22].
(5)⇔(6) Follows from Theorem 4.12 and Corollary 4.15. □
Note that the Z-module Qis a non-finitely generated Z-module that satisfies (1),
(3) and (4) of Theorem 5.1 but fails on (2), (5) and (6). Like for rings, the notions of
(weakly-)morphic and reduced modules connect well to provide conditions related
to regularity in modules. In the subcategory of finitely generated modules, the two
properties combined coincide with different regularity notions in Theorem 5.1. Now
we give a condition in Proposition 5.2 when the endoregular and the strongly F-
regular modules coincide with the modules in Theorem 5.1. Further, we characterize

CHARACTERIZATIONS OF REGULAR MODULES 19
the endoregular and the strongly F-regular modules in terms of (weakly-)morphic
and reduced (sub)modules.
Proposition 5.2. Let Rbe a commutative ring and Mbe a nontrivial finitely
presented R-module. Then the following statements are equivalent:
(1) Mis weakly-morphic and reduced,
(2) R/AnnR(M)is a regular ring,
(3) Mis (weakly-)endoregular,
(4) Mis weakly JT-regular,
(5) Mis (strongly) F-regular,
(6) Every cyclic submodule of Mis a (weakly-)morphic and reduced (resp.,
(weakly-)endoregular, Abelian endoregular, co-reduced, weakly JT-regular,
F-regular) module.
Proof. The equivalence of (2) ⇔(3) ⇔(5) follows from [1, Theorem 23]. The rest
of the equivalences follow from Theorem 5.1. □
Remark 5.3. None of the following notions: Mis “reduced”, “weakly-morphic
+ reduced”, “weakly-morphic + co-reduced” implies S:= EndR(M) is a reduced
ring. Hence, neither weakly JT-regular, (strongly) F-regular, weakly-endoregular
implies Abelian endoregular. There exists a reduced module Mwith every cyclic
submodule weakly-morphic, reduced and co-reduced but with Snot reduced, see
Example 5.4.
Example 5.4. [1, Example 24] Let Rbe a commutative regular ring with a non-
finitely generated maximal ideal M, and let R:= R/Mand M:= RLR. Then M
is a finitely generated strongly F-regular module and therefore, by Lemma 4.8, Mis
weakly-morphic, reduced and co-reduced. However, we claim that S:= EndR(M)
is not a reduced ring. Note that since
S∼
="EndR(R) HomR(¯
R, R)
HomR(R, ¯
R) EndR(¯
R)#∼
="R0
R R #,
the endomorphism φcorresponding to "0 0
1 0 #is non-zero but φ2= 0.
Definition 5.5. Let Rbe a commutative ring. An R-module Mis almost locally
simple module [1] if MMis a trivial or simple RM-module (equivalently if MMis
a trivial or simple R-module) for each maximal ideal Mof R.
It is well known that Ris an almost locally simple R-module if and only if Ris
a regular ring. By Anderson, Chun & Juett in [1, pg. 2], the “almost locally simple
property” in modules is another form of module-theoretic regularity.

20 PHILLY IVAN KIMULI AND DAVID SSEVVIIRI
Lemma 5.6. Let Rbe a commutative ring and Mbe a nontrivial R-module. Then
(1) [1, Theorem 4] Mis JT-regular if and only if Mis a multiplication and
weakly JT-regular module;
(2) [1, Theorem 13] Mis JT-regular ⇒Mis almost locally simple ⇒Mis
strongly F-regular ⇒Mis F-regular ⇒Mis weakly JT-regular.
Corollary 5.7. Let Rbe a commutative ring and Mbe a nontrivial multiplication
R-module. Then Mis JT-regular ⇔Mis almost locally simple ⇔Mis strongly
F-regular ⇔Mis F-regular ⇔Mis weakly JT-regular.
Proof. This is immediate from Lemma 5.6. □
Naoum in [19] proved that a multiplication module is strongly F-regular if and
only if its ring of endomorphisms Sis regular (i.e., Mis endoregular). Proposi-
tion 5.8 shows that for finitely generated multiplication modules over commutative
rings, the (weakly-)morphic and reduced modules coincide with all the regularity
notions we have discussed.
Proposition 5.8. Let Rbe a commutative ring and Mbe a nontrivial finitely
generated multiplication R-module. Then the following statements are equivalent:
(1) Mis (weakly-)morphic and reduced,
(2) R/Ann(M)is a regular ring,
(3) Mis (weakly-)endoregular,
(4) Mis (Abelian) endoregular,
(5) Mis (weakly) JT-regular,
(6) Mis almost locally simple,
(7) Mis (strongly) F-regular,
(8) Every cyclic submodule of Mis a (weakly-)morphic and reduced (resp.,
(weakly-)endoregular, Abelian endoregular, co-reduced, (weakly) JT-regular,
(strongly) F-regular, almost locally simple) module.
Proof. (1)⇔(2) ⇔(3) ⇔(5) ⇔(8) Since weakly-morphic finitely generated multi-
plication modules are morphic by Lemma 3.11, the proof of the equivalence follows
from Theorem 5.1 and Corollary 5.7.
(5)⇔(6) ⇔(7) Follows from Corollary 5.7.
(3)⇔(4) Follows from Proposition 3.13. □
Ware [28, Definition 2.3] calls Mregular (call it W-regular) if it is projective and
every homomorphic image of Mis flat, or equivalently if Mis projective and every
cyclic submodule of Mis a direct summand. To extend the Ware regularity notion,
Zelmanowitz defined the non-projective regular modules. Mis a Zelmanowitz regu-
lar [29] (call it Z-regular) module if given any m∈M, there exists φ∈HomR(M, R)

CHARACTERIZATIONS OF REGULAR MODULES 21
such that mφ(m) = m, or equivalently if for any m∈M, mR is pro jective and is a
direct summand of M.
Remark 5.9. Let Rbe a commutative ring and Mbe a nontrivial R-module.
(a) Since (by [25]) Mis W-regular ⇒Mis Z-regular ⇒Mis strongly F-regular ⇒
Mis F-regular, the W-regular modules and the Z-regular modules are weakly-
endoregular, weakly-morphic and reduced by Lemma 4.8.
(b) If Mis projective, then Mis W-regular ⇔Mis Z-regular ⇔Mis (strongly)
F-regular ⇔for each m∈M, mR is a (weakly-)morphic and reduced mod-
ule ⇔for each m∈M, mR is a co-reduced module ([28, Proposition 2] and
Theorem 4.12).
Acknowledgment. We would like to thank the anonymous referee for the careful
proofreading of our manuscript and for making many valuable comments.
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CHARACTERIZATIONS OF REGULAR MODULES 23
Philly Ivan Kimuli
Department of Mathematics
Faculty of Science
Muni University
P.O. Box 725, Arua, Uganda
e-mail: pi.kimuli@muni.ac.ug, kpikimuli@gmail.com
David Ssevviiri (Corresponding Author)
Department of Mathematics
College of Natural Sciences
Makerere University
P.O. Box 7062, Kampala, Uganda
e-mail: david.ssevviiri@mak.ac.ug, ssevviiridaudi@gmail.com