# On $(\sigma,\delta)$-skew McCoy modules

## TL;DR

This paper introduces and explores the properties of $(\sigma,\delta)$-skew McCoy modules, generalizing existing module concepts and analyzing their behavior in skew triangular matrix extensions.

## Contribution

It defines $(\sigma,\delta)$-skew McCoy modules, establishes their properties, and links them to skew triangular matrix extensions, broadening the understanding of module theory.

## Key findings

- $(\sigma,\delta)$-skew McCoy modules generalize McCoy and $\sigma$-skew McCoy modules.
- Connections between $(\sigma,\delta)$-skew McCoyness and $(\sigma,\delta)$-compatibility are established.
- Characterizations of $(\sigma,\delta)$-skew McCoy modules via skew polynomial extensions are provided.

## Abstract

Let $(\sigma,\delta)$ be a quasi derivation of a ring $R$ and $M_R$ a right $R$-module. In this paper, we introduce the notion of $(\sigma,\delta)$-skew McCoy modules which extends the notion of McCoy modules and $\sigma$-skew McCoy modules. This concept can be regarded also as a generalization of $(\sigma,\delta)$-skew Armendariz modules. Some properties of this concept are established and some connections between $(\sigma,\delta)$-skew McCoyness and $(\sigma,\delta)$-compatible reduced modules are examined. Also, we study the property $(\sigma,\delta)$-skew McCoy of some skew triangular matrix extensions $V_n(M,\sigma)$, for any nonnegative integer $n\geq 2$. As a consequence, we obtain: (1) $M_R$ is $(\sigma,\delta)$-skew McCoy if and only if $M[x]/M[x](x^n)$ is $(\overline{\sigma},\overline{\delta})$-skew McCoy, and (2) $M_R$ is $\sigma$-skew McCoy if and only if $M[x;\sigma]/M[x;\sigma](x^n)$ is $\overline{\sigma}$-skew McCoy.