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We study modules in which perspectivity of summands is transitive. Generalizing a 1977 result of Handelman and a 2014 result of Garg, Grover, and Khurana, we prove that for any ring R, perspectivity is transitive in M 2 (R) if and only if R has stable range one. Also generalizing a 2019 result of Amini, Amini, and Momtahan we prove that a quasi-continuous module in which perspectivity is transitive is perspective. [ABSTRACT FROM AUTHOR]

Let a be a regular element of a ring R. If either K:=r( a) has the exchange property or every power of a is regular, then we prove that for every positive integer n there exist decompositions where $Y_{n} \subseteq a^{n}R$ and E ≅ R/ a R. As applications we get easier proofs of the results that a strongly π-regular ring has stable range one and also that a strongly π-regular element whose every power is regular is unit-regular. [ABSTRACT FROM AUTHOR]

For any element a in an exchange ring R, we show that there is an idempotent $\,e\in aR\cap R\,a\,$ such that $\,1-e\in (1-a)\,R\cap R\,(1-a)$. A closely related result is that a ring R is an exchange ring if and only if, for every a∈ R, there exists an idempotent e∈ R a such that 1− e∈(1− a) R. The Main Theorem of this paper is a general two-sided statement on exchange elements in arbitrary rings which subsumes both of these results. Finally, applications of these results are given to the study of the endomorphism rings of exchange modules. [ABSTRACT FROM AUTHOR]

We show that in a ring of stable range 1, any (von Neumann) regular element is clean. Our main results also imply that any unit-regular ring has idempotent stable range 1 (and is therefore clean), and that a semilocal ring has idempotent stable range 1 if and only if it is semiperfect. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. 'Classroom' is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

Examines the endomorphism rings of the Harada modules. Proof of the Kasch's Theorem using the property of Harada modules; Direct summands of the Harada module; Notations and terminology on the full linear ring.