Communications in Algebra ( IF 0.6 ) Pub Date : 2020-09-18 , DOI: 10.1080/00927872.2020.1813744 Juncheol Han 1 , Yang Lee 2, 3 , Sangwon Park 4
Abstract
We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then for some , by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset . (i) Suppose that ab is central for and ab – ba is a nonzero nilpotent. Then, is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, , and A is the algebra generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for . Then, there exists such that, for every and for all ; especially . We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI.
中文翻译:
与中心有关的NI环的结构
摘要
我们首先获得NI环满足以下属性:如果ab对于元素a,b居中,则 对于一些 ,通过应用缩环的属性。接下来我们证明以下内容:令R为环,而I为子集生成的R的理想值。(i)假设ab对于和AB - BA是一个非零幂零。然后,是R的子环A的非零幂零理想,其中1是R的标识,并且A是代数通过生成一个,b过乙。(ii)如果R为NI,则I为nil,R / I为Abelian NI环。(iii)设R为可逆且a为中心。然后,存在 这样,对于每个 和 对全部 ; 特别。如果它满足要提及的NI环的第一个特性,我们就称其为环伪NI,并在几种环理论情况下检查NI和伪NI环的结构,这表明半简单的Artinian环是伪NI。