On left annihilating content polynomials and power series

Abstract

Let R be an associative unital ring, and let $f\in R[x]$. We say that f is a left annihilating content (AC) polynomial if f = af₁ for some $a\in R$ and $f_1\in R[x]$ with ${\mathcal{l}}_{R[x]}(f_1)=0$. The ring R is called a left EM-ring if each $f\in R[x]$ is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element $a\in R$ and a finitely generated right ideal J of R with ${\mathcal{l}}_R(J)=0$ and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring $R[G]$ is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, $R[x],R[x,x^{-1}]$, and $R[[x]]$ are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings).

KEYWORDS:

• Annihilating content polynomials
• EM-rings
• power series rings
• reversible rings
• right duo rings

2020 MATHEMATICS SUBJECT CLASSIFICATION 16U99:

• 16W60
• 16S36