A Class of Balanced Non-Uniserial Rings.
In this paper, we introduce a subclass of strongly clean rings. Let be a ring with identity, be the Jacobson radical of , and let denote the set of all elements of which are nilpotent in . An element is called very -clean provided that there exists an idempotent such that and or is an element of . A ring is said to be very -clean in case every element in is very -clean. We prove that every very -clean ring is strongly -rad clean and has stable range one. It is shown...
A ring is defined to be left almost Abelian if implies for and , where and stand respectively for the set of idempotents and the set of nilpotents of . Some characterizations and properties of such rings are included. It follows that if is a left almost Abelian ring, then is -regular if and only if is an ideal of and is regular. Moreover it is proved that (1) is an Abelian ring if and only if is a left almost Abelian left idempotent reflexive ring. (2) is strongly...