On (weakly) precious rings associated to central polynomials

Abstract

Let R be an associative ring with identity and let g(x) be a fixed polynomial over the center of R. We define R to be (weakly) g(x)-precious if for every element a∈R, there are a zero s of g(x), a unit u and a nilpotent b such that (a=±s+u+b) a=s+u+b. In this paper, we investigate many examples and properties of (weakly) g(x)-precious rings. If a and b are in the center of R with b-a is a unit, we give a characterizations for (weakly) (x-a)(x-b)-precious rings in terms of (weakly) precious rings. In particular, we prove that if 2 is a unit, then a ring is precious if and only it is weakly precious. Finally, for n∈ℕ, we study (weakly) (xⁿ-x)-precious rings and clarify some of their properties.