相对于幺半群的McCoy环的扩张

Extensions of McCoy Rings Relative to a Monoid

中文关键词:  幺半群  $u.p.$-幺半群  McCoy环  $M$-McCoy环  上三角矩阵环.

英文关键词:monoid  unique product monoid  McCoy ring  $M$-McCoy ring  upper triangular matrix ring.

中文摘要:

      对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环；得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环，则~$R[M]$~是~McCoy~环；对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.

英文摘要:

      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.