Idempotents and completely semiprime ideals
Idempotents and completely semiprime ideals
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DOI：
10.1080/00927878308822865
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发表时间：
1983
影响因子：
0.7
通讯作者：
Gary F. Birkenmeir
中科院分区：
文献类型：
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作者：
Gary F. Birkenmeir
右理想X是稠密零(Dn)的，如果X=0，或者X f 0，且包含在X中的R的每个非零右理想与N 121有非零交。稠密零右理想的概念推广了本质幂零右理想[I]和[%I]的概念，并确定了不包含R的约化右理想的右理想类。右奇异理想Z提供了不一定为零的DN理想的一个例子[A，引理3.31。本文源于关于约化右理想中幂等元的结果，然而，通过定义右半中心幂等元的概念，许多结果将在更一般的背景下成立。幂等元E是右半中心的，如果Er=ere(等价地，对于r EURO R，则Er=ere)。因此，每个幂等元都是右半中心模为幂零元，如果x=x2，则(xr-xrx)2=0。由于约化右理想中的每个幂等元都是右半中心的，所以可以看出极小右理想要么是dN，要么是由右半中心幂等元生成的。如果R的一个幂等元e是右半中心的，或者R是半素的，或者e是左半中心的(即Re=ere)，则e是中心的。引理1从[A]和[GI]总结了右半中心幂等元的基本性质。
COWLETELY SEMIPRIME IDEALS 569 A right ideal X is densely nil (DN) if either: X= 0; or X f 0 and every nonzero right ideal of R which is contained in X has nonzero intersection with N 121. The notion of a densely nil right ideal generalizes that of an essentially nilpotent right ideal [i] and [% I, and determines the class of right ideals which contain no reduced right ideals of R. The right singular ideal Z provides an example of a DN ideal which is not necessarily nil [A, Lemma 3.31.This paper grew out of results on idempotents in reduced right ideals, However it became evident that by defining the notion of a right semicentral idempotent many of the results would hold in a more general context. An idempotent e is right semicentral if eR= eRe (equivalently, for r€ R then er= ere). Hence every idempotent is right semicentral modulo a nilpotent element for if x= x2 then (xr-xrx) 2= 0. Since every idempotent in a reduced right ideal is right semicentral, one can see that a minimal right ideal is either DN or generated by a right semicentral idempotent. Also if an idempotent e of R is right semicentral and either R is semiprime or e is is left semicentral (ie Re= eRe), then e is central. From [A] and [GI the basic properties of a right semicentral idempotent are summarized in Lemma 1.