On a Theorem of Fuller
On a Theorem of Fuller
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关于富勒定理
DOI：
10.1006/jabr.1993.1005
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发表时间：
2009
影响因子：
0.9
通讯作者：
K. Oshiro
中科院分区：
文献类型：
--
作者：
Yoshitomo Baba;K. Oshiro
设R是左Artin环，R中有一个本原幂等元.在[2]中，Fuller证明了以下有用的结果：eRR是单射的当且仅当R中存在本原幂等元f使得（eR，Rf）是1 ′-对，即Socle（eRR）：_/′ R/_fN和Socle（RRf）R2 Re/Ne.其中N是R的Jacobson根。在这种情况下，，鉴于他的证明，富勒使用森田对偶[5]或立川的工作[6]，我们使用“左阿廷”的假设为他的证明。本文给出了定理的一个初等证明，并将定理改进为如下形式：设R是半准素环。然后
Let R be a left artinian ring and ea primitive idempotent in R. In [2], Fuller showed the following useful result: eRR is injective if and only if there exists a primitive idempotent f in R such that (eR, Rf) is an 1'-pair, ie, Socle (eRR): _/‘R/_fN and Socle (RRf) R 2 Re/Ne. where N is the Jacobson radical of R. In this case,,, R/‘is also injective. In view of his proof, Fuller uses Morita duality [5] or Tachikawa’s work [6], and we use the assumption “left artinian” for his proof. The purpose of our paper is to give an elementary proof of the theorem and improve the theorem in the following form: Let R be a semiprimary ring. Then