let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

Nagahara and Kishimoto [i] studied free ring extensions B(x) of degree n n for some integer n over a ring B with i, where x b, cx xp(c) for all c and n-I some b in B (p automorphism of B), and {i, x x is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternlon algebras in which b -I and p is of order 2. The purpose of the pre...

in this work, we investigate the transfer of some homological properties from a ring $r$ to its amalgamated duplication along some ideal $i$ of $r$ $rbowtie i$, and then generate new and original families of rings with these properties.

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...

In this study, we prove that let N be a fixed positive integer and R semiprime -ring with extended centroid . Suppose additive maps such is onto, satisfy one of the following conditions belong to Г-N- generalized strong commutativity preserving for short; (Γ-N-GSCP) on Г-N-anti-generalized (Γ-N-AGSCP) Then there exists an element form when condition (i) satisfied, (ii) satisfied

Twisted generalized Weyl algebras (TGWAs) are a large family of that includes several interest for ring theory and representation theory, such as algebras, primitive quotients $U(\mathfrak{sl}_2)$, multiparameter quantized algebras. In this work, we study invariants TGWAs under diagonal automorphisms. Under certain conditions, able to show the fixed TGWA by an automorphism is again TGWA. partic...

We study the boundary behaviour of open, light mappings satisfying generalized modular inequalities in general metric measure spaces. extend this way known facts from theory quasiregular and their recent generalizations, namely finite distortion so called ring mappings.

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.