## 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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