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A note on centralizers. (English) Zbl 0962.16016
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.
MSC:
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
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