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For a subset S of a ring R, an element a is said to have left S-element stable range 1 if whenever Ra + Rb = R there exists r ∈ S such that a + rb is a unit. When S = R (.resp., S = U (R) and S = idem (R) )., this is the notion of left stable range 1 (.resp., left unit-1 stable range and left idempotent stable range 1). This article is concerned with S = nil (R) . [ABSTRACT FROM AUTHOR]

We give the necessary and sufficient conditions for an n × n matrix over an integral domain to be a sum of involutions and, respectively, a sum of tripotents. We determine the integral domains over which every n × n matrix is a sum of involutions and, respectively, a sum of tripotents. We further determine the commutative reduced rings over which every n × n matrix is a sum of two tripotents. [ABSTRACT FROM AUTHOR]

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2] and were extensively investigated by Ahn and Anderson [1], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings. [ABSTRACT FROM AUTHOR]

AnR-moduleM is called aC4-module if, wheneverA1andA2are submodules ofMwithandis anR-homomorphism with, we have. This notion is a non-trivial generalization of both theC3-modules and the square-free modules. Furthermore, aC4-module is called pseudo-continuous if it is also aCS-module. In this paper, we give a thorough study of theC4- and the pseudo-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]

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]

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that an ≠ 0 and anR = r(l(an)). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries. [ABSTRACT FROM AUTHOR]

Motivated by a result of Mochizuki [Mochizuki, H. (1965). Finitistic global dimension for rings. Pacific J. Math. 15(1):249-258] that, for a left perfect ring R with Jacobson radical J(R), Ext¹R(R/J(R), R) = 0 if and only if R is right self-injective, we prove here that, for a semiperfect ting R with essential right socte Sr, (1) R is tight FP-injective if every R-homomorphism from a finitely generated small subraodule of a free tight R-module F to Sr can be extended to an R-homomorphism from F to R, (2) R is right simple-injective if every R-homomorphism from a small right ideal of R to R with simple image can be extended to an R-homomorphism from R to R, and (3) R is right self-injective if every R-homomorphism from a small right ideal of R to R can be extended to an R-homomorphism from R to R. As consequences, several known results on right self-injective tings, tight FP-injective rings and right simple-injective tings are extended to larger classes of rings. [ABSTRACT FROM AUTHOR]

In this corrigendum, statement (5) of Proposition 4.4 is reformulated and an example is given to show that this correction is necessary. [ABSTRACT FROM AUTHOR]