On (weakly) precious rings associated to central polynomials

Hani A Khashan

doi:10.5269/bspm.v36i2.31398

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.

Highlights

We first determine some conditions under which g(x)-clean rings and g(x)-precious rings are the same
We prove that if 2 ∈ U (R), the statements R is precious and R is weakly (x2 − 1)-precious are equivalent
It is clear that any precious ring is weakly precious

Summary

Introduction

We characterize (weakly) g(x)-precious rings by modules. It is clear that any precious ring is weakly precious. It is clear that every g(x)-clean element in a ring is g(x)-precious. For a non trivial example, one can verify that the ring Z6 is (weakly) (x2 − 2x)-precious which is not (weakly) (x2 − 2x)-clean. [17], every g(x)-nil clean ring is g(x)-precious.

Results

Conclusion