| 相对于幺半群的McCoy环的扩张 |
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| 引用本文： | 杨世洲,宋雪梅. 相对于幺半群的McCoy环的扩张[J]. 数学研究及应用, 2008, 28(3): 659-665. DOI: 10.3770/j.issn.1000-341X.2008.03.031 |
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| 作者姓名： | 杨世洲 宋雪梅 |
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| 作者单位： | 西北师范大学数学与信息科学学院,甘肃 兰州 730070;兰州城市学院数学系,甘肃兰 州 730070 |
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| 基金项目： | 国家自然科学基金(No.60574025); 甘肃省自然科学基金(No.3ZS061-A25-015); 甘肃省教育厅科学基金(Nos.0602-21; 0410B-09). |
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| 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.
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| 关 键 词： | 幺半群 $u.p.$-幺半群 McCoy环 $M$-McCoy环 上三角矩阵环. |
| 收稿时间： | 2006-07-18 |
| 修稿时间： | 2008-03-08 |
Extensions of McCoy Rings Relative to a Monoid |
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| YANG Shi Zhou and SONG Xue Mei. Extensions of McCoy Rings Relative to a Monoid[J]. Journal of Mathematical Research with Applications, 2008, 28(3): 659-665. DOI: 10.3770/j.issn.1000-341X.2008.03.031 |
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| Authors: | YANG Shi Zhou and SONG Xue Mei |
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| Affiliation: | 1. College of Mathematics and Information Science, Northwest Normal University, Gansu 730070, China
2. Department of Mathematics, Lanzhou City University, Gansu 730070, China |
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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 $ntimes n$ 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. |
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| Keywords: | monoid unique product monoid McCoy ring $M$-McCoy ring upper triangular matrix ring. |
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