On Rings Whose Elements are the Sum of a Unit and a Root of a Fixed Polynomial

g.x/-full clean rings

“…Generally, the polynomial ring ROEt is not g.x/-clean for an arbitrary nonzero polynomial g.x/ 2 C.R/OEx. For example let R be a commutative ring, then the polynomial ring ROEt is not g.x/-clean ring [3]. Full elements and invertible elements are the same when the ring R is commutative, so the concept of g.x/-clean and g.x/-full clean are equivalent for commutative rings.…”

“…In Fan and Yang [3], proved that if R is g.x/-clean, then so is M n .R/ for all n 1. Here we have a similar result for g.x/-full clean.…”

“…Following Camillo and Simon [2], an element r 2 R is called g.x/-clean if r D u C s where g.s/ D 0 and u is a unit of R, and R is g.x/-clean if every element is g.x/-clean. Moreover, Fan and Yang have studied g.x/-clean rings and their generalizations in [3]. Ashrafi and Ahmadi also studied weakly g.x/-clean rings [1].…”

ON WEAKLY $g(x)$-NIL CLEAN RINGS

“…Recently, Fan and Yang [12], studied more properties of g(x)-clean rings. Among many conclusions, they proved that if g(x) ∈ (x − a)(x − b)C(R) [x] where a, b ∈ C(R) are such that b − a is a unit in R, then R is a clean ring if and only if R is (x − a)(x − b)-clean.…”

Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute

“…R M / is g.x/-clean. In 2008, Fan and Yang [8], studied more properties of g.x/-clean rings. Among many conclusions, they proved that if g.x/ 2 .x a/.x b/C.R/OEx where a; b 2 C.R/ with b a a unit in R, then R is a clean ring if and only if R is .x a/.x b/-clean.…”