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

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 $n\ge 2$. 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.

13 pages