Weakly clean and related rings / Qua Kiat Tat - Qua, Kiat Tat (2015)
Let R be an associative ring with identity. Let Id(R) and U(R) denote the set of idempotents and the set of units in R, respectively. An element x 2 R is said to be weakly clean if x can be written in the form x = u+e or x = u−e for some u 2 U(R) and e 2 Id(R). If x is represented uniquely in this form, whether x = u + e or x = u − e, then x is said to be uniquely weakly clean. We say that x 2 R is pseudo weakly clean if x can be written in the form x = u+e+(1−e)rx or x = u − e + (1 − e)rx for some u 2 U(R), e 2 Id(R) and r 2 R. For any positive integer n, an element x 2 R is n-weakly clean if x = u1 +· · ·+un +e or x = u1 + · · · + un − e for some u1, . . . ,un 2 U(R) and e 2 Id(R). The ring R is said to be weakly clean (uniquely weakly clean, pseudo weakly clean, n-weakly clean) if all of its elements are weakly clean (uniquely weakly clean, pseudo weakly clean, n-weakly clean). Let g(x) be a polynomial in Z(R)[x] where Z(R) denotes the centre of R. An element r 2 R is g(x)-clean if r = u + s for some u 2 U(R) and s 2 R such that g(s) = 0 in R. The ring R is said to be g(x)-clean if all of its elements are g(x)-clean. In this dissertation we investigate weakly clean and related rings. We determine some characterisations and properties of weakly clean, pseudo weakly clean, uniquely weakly clean, n-weakly clean and g(x)-clean rings for certain types of g(x) 2 Z(R)[x]. Some generalisations of results on clean and related rings are also obtained.