RING BERSIH-g(x) KUAT; ( ON STRONGLY g(x)-CLEAN RINGS )

FRISCA MAREYTA PONGOH, Indah Emilia Wijayanti

2015 | Disertasi | FMIPA

Let R be an associative ring with identity, let C(R) denoted the center of R, and let g(x) be a polynomial in the polynomial ring C(R)[x]. Ring R is called strongly g(x)-clean if every element r 2 R can be written as r = e + u with g(e) = 0, u is a unit of R, and eu = ue. The relation between strongly g(x)-clean rings and strongly clean rings are determined, some general properties of strongly g(x)-clean rings and clean ideal are given. In this paper, by the definition and properties of clean ideal, we introduce definition and properties of g(x)-clean ideal with some examples of it. If ring R is g(x)-clean we must have that g(x) has at least two roots in R. But, for an ideal in ring R, it can be g(x)-clean although g(x) only has one root in R. The example for this case is given.

Kata Kunci : N