| 陈焕艮.环的B-稳定正则理想[J].数学年刊A辑,2012,33(1):123~ |
| 环的B-稳定正则理想 |
| On B-Stably Regular Ideals of a Ring |
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| DOI： |
| 中文关键词: B-稳定理想, 正则理想, 模消去律 |
| 英文关键词:B-Stable ideal, Regular ideal,Cancellation of module |
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| 中文摘要: |
| 研究了正则理想是B-稳定的充分和必要条件, 并且证明环$R$的正则理想$I$是B-稳定的当且仅当对任意的有限生成投射右$R$-模$A$, 如果$A_1$和$A_2$是$A$的有限生成子模且满足 $A_1\cong A_2, A_1=A_1I$ 以及$A_2=A_2I$, 则存在一个有限生成子模$B$, 使得$A=A_1\oplus B=A_2\oplus B$; 当且仅当对任意的幂等元$e, f\in I$, $eR\cong fR$蕴含$eR/(eR\capfR)\cong fR/\big(eR\cap fR)$; 当且仅当对任意的$a\in 1+I$, 存在一个幂等元$e\in I$, 使得$a-e\inU(R)$ 并且$aR\cap eR=0$. 进而构造了相关的例子. |
| 英文摘要: |
| In this paper, the author investigates the necessary and sufficient conditions under which a regularideal is B-stable. It is shown that a regular ideal $I$ of a ring$R$ is B-stable if and only if for any finitely generatedprojective right $R$-module $A$, if $A_1$ and $A_2$ are finitelygenerated submodules of $A$ such that $A_1\cong A_2, A_1=A_1I$ and$A_2=A_2I$, there exists a finitely generated submodule $B$ of $A$, such that $A=A_1\oplus B=A_2\oplus B$; if and only if for anyidempotents $e,f\in I$, $eR\cong fR$ implies $eR/(eR\capfR)\cong fR/(eR\cap fR)$; if and only if for any$a\in 1+I$, there exists an idempotent $e\in I$, such that $a-e\inU(R)$ and $aR\cap eR=0$. Related examples are constructed aswell. |
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