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On center-like elements in rings. (English) Zbl 0573.16009

The results in this paper are based on the following theorem of I. N. Herstein: Let R be a prime ring with center Z, I a nonzero ideal of R, \(a\in R\), and \(n>0\) a fixed integer. If for all \(x\in I\), \((ax-xa)^ n=0\) then \(a\in Z\); and, if \((ax-xa)^ n\in Z\) then either \(a\in Z\) or R is an order in a simple algebra four dimensional over its center [I. N. Herstein, J. Algebra 60, 567-574 (1979; Zbl 0436.16014)]. The authors use this theorem to obtain certain generalizations of it. For example, the conclusions of Herstein’s theorem hold if (ax-xa) is replaced with \([\cdot \cdot \cdot [a,x_ 1],\cdot \cdot \cdot,x_ m]\), for m fixed, where \([x,y]=xy-yx\). They also show that when Z is infinite either \([a,[x_ 1,\cdot \cdot \cdot [x_{m-1},x_ m]\cdot \cdot \cdot]^ n=0\), or \((ax^ k-x^ ma)^ n=0\) and char \(R\neq 2\), forces \(a\in Z\), where again, all \(x_ i\in I\) and k,m,n are fixed.
Reviewer: Ch.Lanski

MSC:

16Rxx Rings with polynomial identity
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)

Citations:

Zbl 0436.16014
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