Left R-module Research Articles

Unifying some classical results on Artinian rings and modules

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.

Pure-Hollow Modules and Pure-Lifting Modules

Let be a commutative ring with identity, and be a unitary left R-module. In this paper we, introduce and study a new class of modules called pure hollow (Pr-hollow) and pure-lifting (Pr-lifting). We give a fundamental, properties of these concept. also, we, introduce some conditions under which the quotient and direct sum of Pr-lifting modules is Pr-lifting.

Weakly 2-Prime Sub-Modules

Let R be a commutative ring containing a unit, and let be a left R-module. We define a proper sub-module N of an R-module M to be a weakly 2-prime sub-module if whenever , then either or . This concept is an expansion of the idea of a weakly 2-prime ideal, where an ideal P of R is said to be a weakly 2-prime ideal if for all implies or . Several characteristics of sub-modules that are weakly 2-prime are taken into account.

Direct Summand of Serial Modules

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules.

A new intersection-graph type for modules

Let R be a ring with unity and M a left R-module. We investigate a new graph associated with M called the simple-intersection graph of M, denoted by G S R ( M ) . Our focus is on exploring the relationship between different algebraic properties of M and properties G S R ( M ) , including connectedness, girth and dominating sets. Our results encompass determining the girth and diameter of GSR(M) and providing insights on the clique and domination numbers.

A necessary and sufficient condition for a direct sum of modules to be distributive

Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S ∩ ( T + U ) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited.

Auslander conditions and tilting-like cotorsion pairs

We study homological behavior of modules satisfying the Auslander condition. Assume that AC is the class of left R-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in AC belongs to AC for any ring R. As a consequence, we show that for any left Noetherian ring R, AC is a resolving subcategory of the category of left R-modules if and only if RR satisfies the Auslander condition if and only if each Gorenstein projective left R-module belongs to AC. As an application, we prove that, for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if AC coincides with the class of Gorenstein projective left R-modules if and only if (AC<∞,(AC<∞)⊥) is a tilting-like cotorsion pair if and only if (AC<∞,I) is a tilting-like cotorsion pair, where AC<∞ is the class of left R-modules with finite AC-dimension and I is the class of injective left R-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.

Left co-Köthe rings and their characterizations

Köthe’s classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe’s problem for basic finite-dimensional algebras. But, so far, Köthe’s problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe’s problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them.

δ-Small Intersection Graphs of Modules

Let R be a commutative ring with unit and M be a unitary left R-module. The δ-small intersection graph of non-trivial submodules of , denoted by , is an undirected simple graph whose vertices are the non-trivial submodules of , and two vertices are adjacent if and only if their intersection is a -small submodule of . In this article, we study the interplay between the algebraic properties of , and the graph properties of such as connectivity, completeness and planarity. Moreover, we determine the exact values of the diameter and girth of , as well as give a formula to compute the clique and domination numbers of

Weakly Quasi 2-Absorbing submodule

Let R be a commutative ring with identity , and M is a unitary left R-module”, “A proper submodule E of an R-module M is called a weakly quasi-prime if whenever r, s ∈ R, m ∈ M, with 0 ≠ rsm ∈ E , implies that rm ∈ E or sm ∈ E”. “We introduce the concept of a weakly quasi 2-absorbing submodule as a generalization of weakly quasi-prime submodule”, where a proper submodule E of M is called a weakly quasi 2-absorbing submodule if whenever r,s,t ∈ R, m ∈M with 0≠ rstm ∈ E , implies that rsm ∈ E or rtm ∈ E or stm ∈ E. we study the basic properties of weakly quasi 2-absorbing. Furthermore, the relationships of weakly quasi 2-absorbing submodule with other classes of module are elistraited.

Generalization of semi-regular modules

We will add new types of modules over a ring ℜ called generalized semi-regular modules. Assuming ℜ is a ring with identity and M is a unitary left ℜ-module which is a duality. We show that our new ideas have characterization and a few properties.

DJ-coessential submodules

Let R be an individual left R-module of the same type as W, with W being a ring containing one. W’s submodules N and K should be referred to as N and K, respectively that K ⊆ N ⊆ W if N/K &lt;&lt;_J (D_j (W)+K)/K, Then K is known as the D J-coessential submodule of Nin W as K⊆_ (Rce) N. Coessential submodule is a generalization of this idea. These submodules have certain interesting qualities, such that if a certain condition is met, the homomorphic image of D J- N has a coessential submodule called D J-coessential submodule.

Weakly Nearly Quasi Prime Submodules

In this paper, all rings are commutative with identity, and all R-modules are unitary Left R-modules. We introduce the concept WNQP submodule as new generalizations of weakly quasi prime submodule and give basic properties, examples and characterizations of this concept.

Ring of Endomorphisms and Modules over a Ring

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ ab ” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab ). 2. A pair of an Abelian group Mab and a ring homomorphism R → ρ R\mathop \to \limits^\rho End (Mab ) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_Mod R (M, N). 4. The set R-module homomorphisms of M to N, denoted by Hom R (M, N), forms R-module. 5. A formal proof of Hom R (¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself as a left R-module. 6. A formal proof of AbGrLMod(M′ ab, ρ′) ≅ M where M′ ab is an abelian group obtained by removing the scalar multiplication from M, and ρ′ is obtained by currying the scalar multiplication of M. The removal of the multiplication from M has been done by the forgettable functor defined as AbGr in the article.

Flow modules and nowhere-zero flows

Let $$\varGamma $$ be a graph, A an abelian group, $${\mathcal {D}}$$ a given orientation of $$\varGamma $$ and R a unital subring of the endomorphism ring of A. It is shown that the set of all mappings $$\varphi $$ from $$E(\varGamma )$$ to A such that $$({\mathcal {D}},\varphi )$$ is an A-flow forms a left R-module. Let $$\varGamma $$ be a union of two subgraphs $$\varGamma _{1}$$ and $$\varGamma _{2}$$ , and $$p^n$$ a prime power. It is proved that $$\varGamma $$ admits a nowhere-zero $$p^n$$ -flow if $$\varGamma _{1}$$ and $$\varGamma _{2}$$ have at most $$p^n-2$$ common edges and both admit nowhere-zero $$p^n$$ -flows. Moreover, it is proved that $$\varGamma $$ admits a nowhere-zero 4-flow if $$\varGamma _{1}$$ and $$\varGamma _{2}$$ both have nowhere-zero 4-flows and their common edges induce a subgraph of size at most 2 or a connected subgraph of size 3. This result can be seen as a generalization of a theorem of Catlin that a graph admits a nowhere-zero 4-flow if it is a union of a cycle of length at most 4 and a subgraph admitting a nowhere-zero 4-flow.

On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.

S-Compressible and s-Prime Modules

Let R be a ring with identity and Ą a left R-module. In this article, we introduce new generalizations of compressible and prime modules, namely s-compressible module and s-prime module. An R-module A is s-compressible if for any nonzero submodule B of A there exists a small f in HomR(A, B). An R-module A is s-prime if for any submodule B of A, annR (B) A is small in A. These concepts and related concepts are studied in as well as many results consist properties and characterizations are obtained.

Homological dimensions relative to preresolving subcategories II

Abstract Let 𝒜 {\mathscr{A}} be an abelian category having enough projective and injective objects, and let 𝒯 {\mathscr{T}} be an additive subcategory of 𝒜 {\mathscr{A}} closed under direct summands. A known assertion is that in a short exact sequence in 𝒜 {\mathscr{A}} , the 𝒯 {\mathscr{T}} -projective (resp. 𝒯 {\mathscr{T}} -injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if 𝒯 {\mathscr{T}} contains all projective (resp. injective) objects of 𝒜 {\mathscr{A}} , then the above assertion holds true if and only if 𝒯 {\mathscr{T}} is resolving (resp. coresolving). As applications, we get that a left and right Noetherian ring R is n-Gorenstein if and only if the Gorenstein projective (resp. injective, flat) dimension of any left R-module is at most n. In addition, in several cases, for a subcategory 𝒞 {\mathscr{C}} of 𝒯 {\mathscr{T}} , we show that the finitistic 𝒞 {\mathscr{C}} -projective and 𝒯 {\mathscr{T}} -projective dimensions of 𝒜 {\mathscr{A}} are identical.

On cofinitely ss-supplemented modules

In this paper, we introduce the concept of (amply) cofinitely ss-supplemented modules as a proper generalization of (amply) ss-supplemented modules, and we provide various properties of these modules. In particular, we prove that arbitrary sum of cofinitely ss-supplemented modules is cofinitely ss-supplemented. Moreover, we show that a ring R is semiperfect and Rad(R)⊆Soc(RR) if and only if every left R-module (amply) cofinitely ss-supplemented.

Characterizing von Neumann Regular Rings in Reverse Mathematics

A von Neumann regular ring, named after John von Neumann for his work related to continuous geometry and operator algebras, has various equivalent characterizations in terms of popular properties of rings and modules. This paper mainly focuses on effective aspects of characterizations of von Neumann regular rings, and studies such characterizations by techniques of reverse mathematics. For not necessarily commutative rings, we obtain that RCA0 proves that a ring R is von Neumann regular iff every Σ10 finitely generated left ideal of R is generated by an idempotent iff every Σ10 cyclic left R-module is Lam-divisible. For commutative rings, we obtain that over RCA0, WKL0 (resp., ACA0) is equivalent to the statement that a commutative ring R is von Neumann regular iff the localization of R at any prime ideal (resp., maximal ideal) exists and it is a field. Lastly, we show that ACA0 proves that a commutative ring R is von Neumann regular iff every simple R-module is Lam-divisible.