J-Boolean like环

J-Boolean Like Ring

本文首先引进了Boolean-like环的一类新的扩张J-Boolean like环,即对任意环R中元素a,b都有(a-a2)(b-b2)∈J(R),这里J(R)为环R的Jacobson根,则环R称为J-Boolean like环.证明了两个定理分别为(1)设D是一个环,C是D的一个子环,R[D,C]是一个J-Boolean like环(a)C,D是J-Boolean like环,(b)J2(C)■J(D).(2)如果B/J(B)是Boolean环,并且B[i]={a+bi|i2=ui+η,a,b,u,η∈B},那么B[i]是J-Boolean like环当且仅当uη∈J(B).

This paper defined a new expansion of Boolean-like rings that was a ring R was called J-Boolean like ring when a, b in any ring R satisfied the condition that （a--a2 ） （b--b2 ）∈ J（R） and J（R） was the Jacobson radical of ring R. Two theorems was also proved. Let D be a ring, C is a subring of D, then RID,C] is a J-Boolean like ring if and only if （a）C,D are J-Boolean like rings, （b）J2 （C）J（D）. If B/J（B） is a Boolean ring, and BEi]= {a＇q-biliz =uiq-v,a,b,η∈B } , then B[＇i] is a J-Boolean like ring if and only if uη∈（B）.