On weak $\alpha$-skew McCoy Rings

TL;DR

This paper introduces weak α-skew McCoy rings, a new generalization of existing ring classes, and explores their properties and relationships with polynomial rings and matrix rings.

Contribution

It defines weak α-skew McCoy rings, establishes their properties, and investigates their connections with matrix rings and polynomial extensions.

Findings

Weak α-skew McCoy rings generalize α-skew McCoy and weak McCoy rings.

Properties of weak α-skew McCoy rings are characterized and analyzed.

Connections between weak α-skew McCoy rings and their polynomial and matrix ring extensions are established.

Abstract

Let $\alpha$ be an endomorphism of a ring $R$. We introduce the notion of weak $\alpha$-skew McCoy rings which are a generalization of the $\alpha$-skew McCoy rings and the weak McCo rings. Some properties of this generalization are established, and connections of properties of a weak $\alpha$-skew McCoy ring $R$ with $n\times n$ upper triangular $T_n(R)$ are investigated. We study relationship between the weak skew McCoy property of a ring $R$ and it's polynomial ring, $R[x]$. Among applications, we show that a number of interesting properties of a weak $\alpha$-skew McCoy ring $R$ such as weak skew McCoy property in a ring $R$.