In this paper, we introduce related comparability for exchange ideals. Let be an exchange ideal of a ring . If satisfies related comparability, then for any regular matrix , there exist left invertible and right invertible such that for idempotents .
We characterize exchange rings having stable range one. An exchange ring has stable range one if and only if for any regular , there exist an and a such that and if and only if for any regular , there exist and such that if and only if for any , .
An exchange ring is strongly separative provided that for all finitely generated projective right -modules and , . We prove that an exchange ring is strongly separative if and only if for any corner of , implies that there exist such that and if and only if for any corner of , implies that there exists a right invertible matrix . The dual assertions are also proved.
A matrix is -clean provided there exists an idempotent such that and . We get a general criterion of -cleanness for the matrix . Under the -stable range condition, it is shown that is -clean iff . As an application, we prove that the -cleanness and unit-regularity for such matrix over a Dedekind domain coincide for all . The analogous for property is also obtained.
Let be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.
It is shown that a ring is a -ring if and only if there exists a complete orthogonal set of idempotents such that all are -rings. We also investigate -rings for Morita contexts, module extensions and power series rings.
In this paper, we prove that unit ideal-stable range condition is right and left symmetric.
In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ; where is a Boolean ring; local ring with nil Jacobson radical; or ; or the ring of a Morita context with zero pairings where the underlying rings are or .
A ring is (weakly) nil clean provided that every element in is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let be abelian, and let . We prove that is nil clean if and only if is Boolean and is nil. Furthermore, we prove that is weakly nil clean if and only if is periodic; is , or where is a Boolean ring, and that is weakly nil clean if and only if is nil clean for all .
A -ring is strongly 2-nil--clean if every element in is the sum of two projections and a nilpotent that commute. Fundamental properties of such -rings are obtained. We prove that a -ring is strongly 2-nil--clean if and only if for all , is strongly nil--clean, if and only if for any there exists a -tripotent such that is nilpotent and , if and only if is a strongly -clean SN ring, if and only if is abelian, is nil and is -tripotent. Furthermore, we explore the structure...
We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a matrix over a projective-free ring is strongly -clean if and only if , or , or is similar to , where , , and the equation has a root in and a root in . We further prove that is strongly -clean if be optimally -clean.
We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).
We introduce a class of rings which is a generalization of reflexive rings and -reversible rings. Let be a ring with identity and denote the Jacobson radical of . A ring is called -reflexive if for any , implies . We give some characterizations of a -reflexive ring. We prove that some results of reflexive rings can be extended to -reflexive rings for this general setting. We conclude some relations between -reflexive rings and some related rings. We investigate some extensions of...
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