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Motivated by Hirano-Tominaga's work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices. [ABSTRACT FROM AUTHOR]

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

A module is called a -module if, whenever and are submodules of with and is a homomorphism with , we have . The class of -modules contains the -modules as well as the dual-square-free (DSF) modules. Furthermore, a -module is called pseudo-discrete if is also a lifting module. In this paper, we study the -, the DSF, and the pseudo-discrete modules, and show that a pseudo-discrete module is clean iff it has the finite exchange property iff it has the full exchange property. [ABSTRACT FROM AUTHOR]

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or . Here we carry out a study of the rings whose cyclic modules are -modules. The motivation is the observation that a ring is semisimple artinian if and only if every -generated right -module is a -module. Many basic properties are obtained for the rings whose cyclics are -modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a matrix ring over a strongly regular ring. Applications to the rings whose -generated modules are -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed. [ABSTRACT FROM AUTHOR]

It is well known that every uniquely clean ring is strongly clean. In this article, we investigate the question of when this result holds element-wise. We first construct an example showing that uniquely clean elements need not be strongly clean. However, in case every corner ring is clean the uniquely clean elements are strongly clean. Further, we classify the set of uniquely clean elements for various classes of rings, including semiperfect rings, unit-regular rings, and endomorphism rings of continuous modules. [ABSTRACT FROM AUTHOR]

This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in [22] are answered. [ABSTRACT FROM PUBLISHER]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]

A module is defined to be an automorphism-invariant module if it is invariant under automorphisms of its injective hull. Quasi-injective modules and, more generally, pseudo-injective modules are all automorphism-invariant. Here we develop basic properties of these modules, and discuss when an automorphism-invariant module is quasi-injective or injective. Some known results on quasi-injective and pseudo-injective modules are extended. [ABSTRACT FROM AUTHOR]

For a polynomialwith zero constant term, a semiprime K-algebra R is called faithful f-free if every nonzero ideal of R does not satisfy f. We prove that a semiprime algebra has an essential ideal which is the direct sum of its largest faithful f-free ideal and its largest ideal satisfying the identity f. Here, faithful f-free algebras are characterized, and applications to some interesting differential identities are discussed. Especially with f = S2n(n ≥ 1), a description of semiprime rings is obtained in terms of faithful S2n-free rings and semiprime PI-rings of degree 2n. Semiprime PI-rings of degree 2n are realized through faithful S2n-rings. Finally, faithful S2n-rings are characterized. [ABSTRACT FROM PUBLISHER]

This article is a continuation of the work in [3] on clean index of rings. Motivated by recent work [5] on uniquely clean rings, the clean index of a ring was introduced in [3] to study the structure of rings, where arbitrary rings of clean indices 1, 2, and 3, respectively, are characterized. In particular, clean rings of clean indices 1, 2, and 3 are determined. In this article, we characterize the rings of clean index 4. As applications, clean rings of clean indices 4, 5, 6 and 7 are completely determined. [ABSTRACT FROM PUBLISHER]

A ring R is said to satisfy the Goodearl-Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x -- u and v --u-1 are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl-Menal condition if and only if no homomorphic image of R is isomorphic to either Z2 or Z3 or M2(Z2). These results correct two existing results. Some consequences are discussed. [ABSTRACT FROM AUTHOR]

Abstract: Let K s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K s(R) to be strongly clean and, respectively, for the ring K s(R) to be strongly clean. Specializing to s =1 yields some known results. New families of strongly clean rings are presented. [Copyright &y& Elsevier]

Abstract: A ring is called a left p.p. ring if every principal left ideal is projective. The objective here is to completely determine the left structural matrix rings over a von Neumann regular ring. [Copyright &y& Elsevier]

A right ideal A of a ring R is called annihilator-small if A+T=R, T a right ideal, implies that $\mathcal{I}\, (T) \, = \, 0$, where $\mathcal{I}$ indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it is annihilator-small are given, its relationship to the total of the ring is examined, and its connection with related rings is investigated. [ABSTRACT FROM AUTHOR]

A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L. Vaš, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388-3400]. Here it is proved that a *-ring R is strongly *-clean if and only if R is an abelian, *-clean ring if and only if R is a clean ring such that every idempotent is a projection. As consequences, various examples of strongly *-clean rings are constructed and, in particular, two questions raised in [L. Vaš, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388-3400] are answered. [ABSTRACT FROM AUTHOR]

Abstract: This is a continuation of recent work on the morphic property of rings. The main objective of this article is to study the relationships between regular rings and quasi-morphic rings, between unit-regular rings and morphic rings, and between strongly regular rings and centrally morphic rings. [Copyright &y& Elsevier]

Abstract: Let M and N be two modules over a ring R. Recent works by Kasch, Schneider, Beidar, Mader, and others have shown that some of the ring and module theory can be carried out in the context of . The study of substructures of such as the radical, the singular and co-singular ideals and the total has raised new questions for research in this area. This paper is a continuation of study of these substructures, focusing on when the total is equal to the radical, as well as their connections with (semi)regularity of . New results obtained include necessary and sufficient conditions for the total to equal the radical, a description of the maximal regular sub-bimodule of , the existence of the maximal semiregular ideal of a ring, and answers to a number of existing open questions. [Copyright &y& Elsevier]

Abstract: We prove a Wedderburn–Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599–613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring. [Copyright &y& Elsevier]

Abstract: A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer , the paper deals with the question whether the strongly clean property of , , and follows from the strongly clean property of . This is ‘Yes’ if by a known result. [Copyright &y& Elsevier]

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean. [ABSTRACT FROM AUTHOR]

Abstract: A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the matrix ring is strongly clean where is the ring of p-adic integers, but is not strongly clean where is the localization of at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for to be strongly clean. As consequences, is strongly clean iff is strongly clean iff is strongly clean iff is strongly clean. [Copyright &y& Elsevier]