| Extensions of McCoy Rings Relative to a Monoid |
| Received:July 18, 2006 Revised:March 08, 2008 |
| Key Words:
monoid unique product monoid McCoy ring $M$-McCoy ring upper triangular matrix ring.
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| Fund Project:the National Natural Science Foundation of China (No.\,10171082); the Natural Science Foundation of Gansu Province (No.\,3ZSA061-A25-015) and the Scientific Research Fund of Gansu Provincial Education Department (Nos.\,06021-21; 0410B-09). |
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| 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.03.028 |
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