A note on centralizers. (English) Zbl 0962.16016

Bell, Howard E.

Int. J. Math. Math. Sci. 24, No. 1, 55-57 (2000).

Let \(R\) be a semiprime ring with center \(Z\), \(V\) be right ideal of \(R\), \([V,V]=\{[a,b];\;a,b\in V\}\) and \(C_R([V,V])\) be the centralizer of \([V,V]\) in \(R\). The author proves the following results: (i) if \(R\) is a prime ring and \(H=C_R([V,V])\cap V\) then \(H\) is a commutative subring of \(R\) and \(H=V\cap Z\), or \(H\) is a zero ring and \(H=\text{Ann}([V,V])\cap V\); (ii) if \(R\) is a prime ring and \([V,V]\) is finite, then \(R\) is either finite or commutative; (iii) if \(V\) has a finite index in \(R\) and \([V,V]\) is finite, then either \(R\) is finite or \(R\) contains a nonzero central ideal.

Reviewer: Yu.N.Mal’tsev (Barnaul)

MSC:

16N60	Prime and semiprime associative rings
16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)

Keywords:

finite rings; commutativity theorems; semiprime rings; centers; centralizers; prime rings; commutative subrings; central ideals