On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

Simple-direct-modules

A right R-module M is called simple-direct-injective if, whenever, A and B are simple submodules of M with A?B, and B?^⊕M, then A?^⊕M. Dually, M is called simple-direct-projective if, whenever, A and B are submodules of M with M∕A?B?^⊕M and B simple, then A?^⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ring R is uniserial (artinian serial) with J²(R) = 0 iff every simple-direct-projective right R-module is an SSP-module (SIP-module) iff every simple-direct-injective right R-module is an SIP-module (SSP-module).     

Simple-Direct-Projective Modules

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a right R-module M is called simple-direct-projective if, whenever A and B are submodules of M with B simple and M/A ? B ?^⊕M, then A ?^⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ring R is artinian and serial with J²(R) = 0 if and only if every simple-direct-projective right R-module is quasi-projective if and only if every simple-direct-projective right R -module is a D3-module. It is also shown that a ring R is uniserial with J²(R) = 0 if and only if every simple-direct-projective right R-module is a C3-module if and only if every simple-direct-injective right R -module is a D3-module.     

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A₁ ⊕ A₂ with M′ ? M, there is a decomposition M′ =M₁ ⊕ M₂ such that A = M′ ⊕ [A₁ ∩ (M₁ ⊕ B)] ⊕ [A₂ ∩ (M₂ ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.     

Separativity of Regular Rings in Which 2 Is Invertible

A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A? A⊕ B? A⊕ B implies that A? B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 ? a ²)R = Rr(a) = ?(a)R and i(End _R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 ? a ²)R ∩ RaR = Rr(a) ∩ ?(a)R ∩ RaR and i(End _R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

On Strong Goldie Dimension

Let R be an associative ring with identity. A unital right R-module M is called “strongly finite dimensional” if Sup{G.dim (M/N) | N ≤ M} < +∞, where G.dim denotes the Goldie dimension of a module. Properties of strongly finite dimensional modules are explored. It is also proved that: (1) If R is left F-injective and semilocal, then R is left finite dimensional. (2) R is right artinian if and only if R is right strongly finite dimensional and right semiartinian. Some known results are obtained as corollaries.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Utumi modules

A right R-module M is called a U-module if, whenever A and B are submodules of M with A?B and A ∩ B = 0, there exist two summands K and L of M such that A?^essK, B?^essL and K⊕L?^⊕M. The class of U-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class of U-modules inherits some of the important features of the aforementioned classes of modules. For example, a U-module M is clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly clean U-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly clean U-module has stable range 1.     

A Note on Range-Inclusive Maps in Rings with Idempotents

Let A be a ring, let M be an A-bimodule, and let C be the center of M. A map F: A → M is said to be range-inclusive if [F(x), A] ? [x, M] for every x ? A. Recently, Bre?ar proved that if A is a unital ring and M a unital A-bimodule such that A contains wide idempotents, then every range-inclusive additive map F: A → M is of the form F(x) = λx + μ(x) for some λ ?C and μ: A → C. Our main purpose is to remove the assumption of unitality in the above result.     

Strongly Duo Modules and Rings

An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M _R . Several equivalent conditions to being strongly duo are given. If M _R is strongly duo and reduced, then End _R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M _R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M _R is strongly duo if and only if either M ≈ R or M _R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R _R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R?I | xy ∈ I for some y ∈ R?I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Characterization of Lie Derivations on Prime Rings

Let ? be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ? satisfies L([A, B]) = [L(A), B] + [A, L(B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L(A) = ?(A) + h(A) for all A ∈ ?, where ? is an additive derivation on ? and h is an additive map into its center.