Characterizations of regular modules

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.


Introduction
Let R be an associative and unital ring that is not necessarily commutative and M be a right Rmodule. We call R (unit-)regular if for each a ∈ R there exists a (unit) y ∈ R such that a = aya. It is strongly regular if for each a ∈ R there exists an element y ∈ R such that a = a 2 y, 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. R is 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 = a 2 R for each a ∈ R. Where R is commutative, it is regular ⇔ it is strongly regular.
Following the (von-Neumann) regularity characterizations for rings, different authors have come up with different definitions for the notion of "regularity" for modules. We outline some of them below (see also Definition 5.1, [28,Definition 2.3] and [29]): Definition 1.1. An R-module M is said to be (a) endoregular [15,28] if ϕ(M) and ker(ϕ) are direct summands of M for every endomorphism ϕ of M; (b) Abelian endoregular [15] if End R (M) is a strongly regular ring; (c) F-regular [8] if for every submodule N of M, the sequence 0 → N E → M E is exact for each R-module E; 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 1); the class of Abelian endoregular modules coincides with that of morphic modules with reduced rings of endomorphisms (Theorem 2); if a module M is strongly F-regular, then each of its sub-module is invariant under every endomorphism of M if and only if M is a morphic module with a reduced ring of endomorphisms (Theorem 3). A module is F-regular if and only if each of its (cyclic) sub-modules is a weakly-morphic and reduced module (Theorem 4). 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 following coincide: weakly-morphic and reduced ⇔ F-regular ⇔ weakly-endoregular ⇔ weakly JT-regular (Theorem 7).
Notation and conventions. Throughout this paper, all rings R will be associative and unital but not necessarily commutative, M is a unitary right R-module and S denotes End R (M), the ring of endomorphisms of M. Therefore, in this case M can be viewed as a left S-right Rbimodule. By Z, Q and R we 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(resp., N ⊆ M) means that N is a submodule (resp., a direct summand) of M. We For any a ∈ R, the principal ideal generated by a is denoted by (a).
The following definitions are necessary in the remaining part of this section and will be used freely in the next sections. (c) said to have Insertion-of-Factors-Property (IFP) if for a, b ∈ R, ab = 0 implies that arb = 0 for every r ∈ R. (c) said to possess IFP if whenever a ∈ R and m ∈ M satisfy ma = 0, then mra = 0 for each element r of R.
The notions in the Definitions 1.2 and 1.3 have been widely studied in [6,9,14,16,17]. A module M R is said to be rigid [6] if given a ∈ R and m ∈ M, the condition ma 2 = 0 implies ma = 0. This is equivalent to l M (a n ) = l M (a) for every a ∈ R and n ∈ Z + . For commutative rings R, it was shown in [14] that M is reduced if and only if l M (a n ) = l M (a) for every a ∈ R and n ∈ Z + . As a dual notion to reduced modules in [14], we have co-reduced modules.
Definition 1.4. Let R be a commutative ring. An R-module M is said to be co-reduced if Ma = Ma n for every a ∈ R and n ∈ Z + .
For noncommutative rings, we give characterizations of reduced modules and reduced rings.
Lemma 1. Let R be a ring and M be a nontrivial R-module. The following statements are equivalent: (1) M is reduced; (2) M is symmetric and l M (a n ) = l M (a) for every a ∈ R and n ∈ Z + ; (3) M has IFP and l M (a n ) = l M (a) for every a ∈ R and n ∈ Z + . Proof.
(1)⇔(2) Assume that (1) holds. By [9, Theorem 2.2], reduced modules are symmetric. To prove that l M (a n ) = l M (a), let x ∈ l M (a n ). Then xa n = 0. As M is reduced, xRa = 0 and so xa = 0. This gives l M (a n ) ⊆ l M (a). Since the reverse inclusion is trivial, we obtain l M (a n ) = l M (a). The proof of (2) ⇒ (1) holds after applying [ (1) R is reduced, (2) l R (a n ) = l R (a) for every a ∈ R and n ∈ Z + , (3) r R (a n ) = r R (a) for every a ∈ R and n ∈ Z + .
If M R is a reduced module over a commutative ring R, then Ma ∼ = Ma n for each a ∈ R and n ∈ Z + . To see this, assume that M is reduced. By Lemma 1, l M (a n ) = l M (a), and so Ma ∼ = M/l M (a) = M/l M (a n ) ∼ = Ma n . For a not necessarily commutative ring R, the map ϕ : M → M given by m → ma for a ∈ R need not be an endomorphism. We show in Proposition 1 that when M is reduced and e is an idempotent element of R, then m → me is an idempotent endomorphism of M.  (1) M is weakly-morphic and reduced, (2) M is weakly-morphic and co-reduced, (3) M is co-reduced and reduced, Proof.
(1)⇒(2) Assume that (1) holds. We need to show that Ma = Ma n for every a ∈ R and n ∈ Z + . Since (2)⇒(1) Assume that (2) holds. Then Ma = Ma n for every a ∈ R and n ∈ Z + . Since M is weaklymorphic, l M (a) ∼ = M/Ma = M/Ma n ∼ = l M (a n ). This gives l M (a) ∼ = l M (a n ). In view of [14,Proposition 2.3], it remains to prove that this is equality. By [13,Lemma 1], there exists some ϕ ∈ S such that l M (a n ) = ϕ(M) and ker(ϕ) = Ma n . The two equalities imply that 0 = ϕ(M)a n = ϕ(Ma n ). By hypothesis, Ma = Ma n . So ϕ(Ma n ) = ϕ(Ma) = ϕ(M)a, from which we deduce that l M (a n ) = ϕ(M) ⊆ l M (a). Hence l M (a n ) = l M (a).
(2)⇒(3) Follows from the proof of (2) ⇒ (1). Corollary 2. Let R be a commutative ring and M be a nontrivial finitely generated R-module. Then the following statements are equivalent: (1) M is weakly-morphic and reduced, (2) M is co-reduced, Proof.
Corollary 3. The following statements are equivalent for a commutative ring R: (1) R is morphic and reduced, Proof.
(2)⇔(3) A commutative ring R is regular if and only if for each a ∈ R, aR = a 2 R. Thus R is co-reduced if and only if it is regular. The equivalences (1)⇔(3)⇔(4) follow from Theorem 1.

Abelian endoregular modules
The focus of this section is the characterization of Abelian endoregular modules in terms of reduced and morphic modules. A ring R is said to be Abelian if all its idempotents are central. If R is reduced, then every idempotent is central. A strongly regular ring is reduced, regular and Abelian. More generally, an R-module M is said to be Abelian if S is an Abelian ring. M R is an Abelian endoregular module if S is a regular and Abelian ring.
Remark 1. M R is an Abelian endoregular module if and only if M = ϕ(M) ker(ϕ) for every ϕ ∈ S. Abelian endoregular modules are morphic modules. Note that an endoregular module need not be morphic.
Lemma 2. If R is a ring and M a nontrivial morphic R-module, then for every ϕ ∈ S, Moreover, the following statements are equivalent: Lemma 3. If M is a morphic R-module and S is a reduced ring, then for every ϕ ∈ S, Proof. We only prove r M (ϕ 2 ) ⊆ r M (ϕ) since the reverse inclusion is obvious. Since M is morphic, there exists γ ∈ S such that γ(M) = r M (ϕ 2 ). This implies ϕ 2 γ = 0. Further, S being reduced implies that ϕγ = 0. So, γ(M) ⊆ r M (ϕ) and we get r M (ϕ 2 ) = γ(M) ⊆ r M (ϕ).
Now we give a characterization of Abelian endoregular modules in terms of morphic modules and reduced rings of endomorphisms.
Theorem 2. Let R be a ring and M be a nontrivial R-module. The following statements are equivalent: (1) M R is a morphic module and S is a reduced ring, Proof. (2)⇒(1) M is morphic by Remark 1. In addition, since S is a strongly regular ring, it is reduced.
Corollary 5. Let R be a ring and M a nontrivial R-module. The following statements are equivalent: (1) M R is morphic and S M is reduced, (2) M R is an Abelian endoregular module.
Proof.  (1) R is right morphic and reduced, (2) R is left P -injective and reduced, (4) R is strongly regular. Proof.
(2)⇒(4) Since R is reduced, for each a ∈ R, l R (a) = l R (a 2 ) by Corollary 1. It follows that aR = r R (l R (a)) = r R (l R (a 2 )) = a 2 R because R is left P -injective. Thus a = a 2 y for some y ∈ R, and this proves R is strongly regular.
A module M R is duo provided every sub-module of M is fully invariant, that is, for any sub-module N of M, ϕ(N) ⊆ N for every ϕ ∈ S. A ring R is right duo if and only if every right ideal of R is a two-sided ideal, equivalently if Ra is contained in aR for every element a in R [24]. (1) M is reduced as a right R-module, (2) M is reduced as a left S-module, (3) S is a reduced ring. Proof.
(1)⇒(2) Let ϕ ∈ S and m ∈ M such that ϕ 2 (m) = 0. Then ma 2 = 0 for some a ∈ R because M is duo. By (1), mra = 0 for all r ∈ R. Since every element in S is defined by right multiplication of each element of M by some element of R, ϕ(ψ(m)) = mra = 0 for every ψ ∈ S for some r ∈ R. Thus ϕS(m) = 0 and S M is a reduced module.
Note that even when a duo module has a reduced ring of endomorphisms, the module itself may not be reduced. Proof. Since every cyclic R-module is a multiplication module that is finitely generated, it is weaklymorphic if and only if it is morphic by Lemma 6.

Proposition 2. Let R be a commutative ring and M be a nontrivial finitely generated multiplication R-module. Then M is weakly-endoregular if and only if it is Abelian endoregular.
Proof. Assume that M R is a weakly-endoregular module. By Theorem 1, M R is weakly-morphic and reduced. As it is a finitely generated multiplication module, Lemma 6 implies M R is morphic. Being duo, S M is a reduced module by Lemma 5. Applying Corollary 5 proves that M is Abelian endoregular. The converse clearly holds since every Abelian endoregular module over a commutative ring is weakly-endoregular.

Corollary 8. Every cyclic module over a commutative ring R is 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 2.
An R-module M is strongly duo [12] if the trace of M in N is N, that is, Tr N (M) := {Im(λ) : λ ∈ Hom R (M, N)} = N for all N ⊆ M R . Clearly, every strongly duo module M is a duo module.
In [12,Theorem 5.5], the ring of endomorphisms of a module M that is strongly duo and reduced was shown to be a strongly regular ring. For commutative rings, we have an improved result in Corollary 9.
Corollary 9. Let R be a commutative ring and M be a nontrivial duo R-module. The following statements are equivalent: (1) M R is a morphic and reduced module, (2) S is a strongly regular ring. Proof.

F-regular modules
Recall that a ring R is regular if and only if every right (left) cyclic ideal of R is a direct summand of R R . To generalise this characterization to modules, Ramamurthi and Rangaswamy in [25, pg. 246] defined strongly regular modules. A module M is 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.

Proposition 4. Every nontrivial strongly F-regular module M R is a k-local-retractable module.
Proof. Since strongly F-regular modules are self-generator modules by [20, pg. 228], M R is P-flat over S by [20,Lemma 1], which is equivalent to being k-local-retractable by [15, pg. 4069].

Lemma 9. Let M be a nontrivial duo and strongly F-regular
Proof. First, we prove that for every submodule N of M, ϕ(N) ⊆ N for all homomorphisms ϕ : N → M. Let n ∈ N and consider ϕ : nR → M. By the strongly F-regular hypothesis, M = nR X for some submodule X. Define β : M → M by β(s + x) = ϕ(s) for every s ∈ nR and x ∈ X.
Then β is a well-defined endomorphism of M which extends ϕ to an endomorphism of M. It follows that for any n ∈ N there exists β ∈ S such that ϕ(n) ∈ ϕ(nR) = β(nR) ⊆ N because M is duo. Hence ϕ(N) ⊆ N. Therefore, if σ : K → K ′ is the given isomorphism, then K ′ = σ(K) ⊆ K and 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 3. Let R be a ring and M be a nontrivial strongly F-regular module. Then the following statements are equivalent: (1) M R is a morphic module and S is a reduced ring, M R is an Abelian endoregular module. Proof.
(1)⇒(2) Assume that (1) holds. Let N be a submodule of M and ϕ ∈ S. By the strongly F-regular hypothesis, for every n ∈ N, nR = e(M) for some idempotent e ∈ S. Since S is reduced, e is central in S. Hence, ϕ(n) ∈ ϕ(nR) = ϕ(e(M)) = e(ϕ(M)) ⊆ e(M) = nR ⊆ N. This proves that ϕ(N) ⊆ N for all ϕ ∈ S, so M R is duo.   By [8, Proposition 8.1], every pure submodule is also RD-pure. A ring R is regular if and only if every (right) ideal is pure (see [8]). Using this fact, Fieldhouse calls M R a regular module if every submodule N of M is pure. Following Naoum [19], we call the Fieldhouse regular modules F -regular. Example 4.1. The converse of Lemma 10 does not hold in general. The Z-module Q is weaklyendoregular, weakly-morphic and reduced but it is not F-regular. In particular, not all its submodules are (RD-)pure since 2Q ∩ Z = 2Z for 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 R is 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 R n → R m → M with n, m ∈ Z + , or equivalently if M ∼ = P/Q, where P and Q are finitely generated modules, and P is a projective module. Clearly, strongly F-regular modules are F-regular but the converse is not true in general, see [1]. In Proposition 5, we determine when the F-regular modules are strongly F-regular. Proof.
(4)⇒(5) Since every cyclic submodule of M is a finitely generated R-module, the proof follows from Corollary 2.
(c) Co-reduced (= Weakly-morphic + reduced) on mR, m ∈ M ⇒ M/mR is finitely presented. Proof. Since weakly JT-regular modules are exactly the co-reduced modules, the proof follows by Theorem 1.

Coincidence of morphic, reduced and regular modules
This section gives conditions under which the different regularity notions of modules coincide with weakly-morphic and reduced modules. Further, under some special 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  (1) M is weakly-morphic and reduced, M is weakly-endoregular, (4) M is weakly JT-regular, Every cyclic submodule of M is a (weakly-)morphic and reduced (resp., weakly-endoregular, Abelian endoregular, co-reduced, weakly JT-regular, F-regular) module. Proof.
(1)⇔(2) ⇔ (4) Follows Corollary 2 and Remark 4 (b) respectively.  Note that the Z-module Q is a non-finitely generated Z-module that satisfies (1), (3) and (4) of Theorem 5 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. Now we give a condition in Proposition 6 when the endoregular and the strongly Fregular modules coincide with the modules in Theorem 5. Further, we characterise the endoregular and the strongly F-regular modules in terms of (weakly-)morphic and reduced (sub)modules.
Proposition 6. Let R be a commutative ring and M be a nontrivial finitely presented R-module. Then the following statements are equivalent: (1) M is weakly-morphic and reduced, (2) R/Ann R (M) is a regular ring, Proof. The equivalence of (2) ⇔ (3) ⇔ (5) follows from [1,Theorem 23]. The rest of the equivalences follow from Theorem 5.
Remark 5. None of the following notions: M is "reduced", "weakly-morphic + reduced", "weaklymorphic + co-reduced" implies S := End R (M) is a reduced ring. Hence, neither weakly JT-regular, (strongly) F-regular, weakly-endoregular implies Abelian endoregular. There exists a reduced module M with every cyclic submodule weakly-morphic, reduced and co-reduced but with S not reduced, see Example 5.1.
Example 5.1. [1, Example 24] Let R be a commutative regular ring with a non-finitely generated maximal ideal M, and let R := R/M and M := R R. Then M is a finitely generated strongly F-regular module and therefore, by Lemma 10, M is weakly-morphic, reduced and co-reduced. However, we claim that S := End R (M) is not a reduced ring. Note that since S ∼ = End R (R) Hom R (R, R) Hom R (R,R) End R (R) ∼ = R 0 R R , the endomorphism ϕ corresponding to 0 0 1 0 is non-zero but ϕ 2 = 0.
Definition 5.1. Let R be a commutative ring. An R-module M is almost locally simple module [1] if M M is a trivial or simple R M -module (equivalently if M M is a trivial or simple R-module) for each maximal ideal M of R.
It is well known that R is an almost locally simple R-module if and only if R is 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.