A combinatorial commutativity property for rings.
Let be fixed positive integers, and let be a ring with unity in which for every in there exist integers such that either or for all . In the present paper it is shown that is commutative if it satisfies the property (i.e. for all implies ).
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...
Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = : ∃ 0 ≤ n∈ ℤ such that , ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring...