Linearly McCoy Rings and Their Generalizations

Abstract

A ring $R$ is called linearly McCoy if whenever linear polynomials $f(x)$, $g(x) ∈ R[x]$\{0} satisfy $f(x)g(x) = 0$, then there exist nonzero elements $r, s ∈ R$ such that $f(x)r = sg(x) = 0$. For a ring endomorphism $α$, we introduced the notion of $α$-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring $R[x; α]$ in place of the ring $R[x]$. A number of properties of this generalization are established and extension properties of $α$-skew linearly McCoy rings are given.