We introduce the notion ofstrongly $alpha$-reversible rings which is a strong version of$alpha$-reversible rings, and investigate its properties. We firstgive an example to show that strongly reversible rings need not bestrongly $alpha$-reversible. We next argue about the strong$alpha$-reversibility of some kinds of extensions. A number ofproperties of this version are established. It is shown ...

It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...

The concept of strongly central reversible rings has been introduced in this paper. It has been shown that the class of strongly central reversible rings properly contains the class of strongly reversible rings and is properly contained in the class of central reversible rings. Various properties of the above-mentioned rings have been investigated. The concept of strongly central semicommutativ...

An element is known a strongly nil* clean if a=e1 - e1e2 + n , where e1,e2 are idempotents and nilpotent, that commute with one another. ideal I of ring R called each element. We investigate some its fundamental features, as well relationship to the nil ideal.

A ring R is uniquely (nil) clean in case for any $a in R$ there exists a uniquely idempotent $ein R$ such that $a-e$ is invertible (nilpotent). Let $C =(A V W B)$ be the Morita Context ring. We determine conditions under which the rings $A,B$ are uniquely (nil) clean. Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.