Fisher, Joe W.; Fahmy, Mohamed H. On center-like elements in rings. (English) Zbl 0573.16009 Int. J. Math. Math. Sci. 8, 145-149 (1985). 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.) Keywords:identities; commutativity; prime ring; center; order in a simple algebra Citations:Zbl 0436.16014 PDFBibTeX XMLCite \textit{J. W. Fisher} and \textit{M. H. Fahmy}, Int. J. Math. Math. Sci. 8, 145--149 (1985; Zbl 0573.16009) Full Text: DOI EuDML