Extensions of McCoy Rings Relative to a Monoid

Abstract:

      For a monoid $M$, we introduce $M$-McCoy rings, which are generalization of McCoy rings, and we investigate their properties. Every $M$-Armendariz ring is $M$-McCoy for any monoid $M$. We show that $R$ is an $M$-McCoy ring if and only if an $n\times n$ upper triangular matrix ring $aUT_n(R)$ over $R$ is an $M$-McCoy ring for any monoid $M$. It is proved that if $R$ is McCoy and $R[x]$ is $M$-McCoy, then $R[M]$ is McCoy for any monoid $M$. Moreover, we prove that if $R$ is $M$-McCoy, then $R[M]$ and $R[x]$ are $M$-McCoy for a commutative and cancellative monoid $M$ that contains an infinite cyclic submonoid.