Commutativity conditions for rings: 1950–2005

Abstract Let R be a 2-torsion free semiprime ring with the centre Z(R), U be a non-zero ideal and d: R → R be a derivation mapping. Suppose that R admits (1) a derivation d satisfying one of the following conditions: (i) [d(x), d(y)] − [x, y] ∈ Z(R) for all x, y ∈ U, (ii) [d2(x), d2(y)] − [x, y] ∈ Z(R) for all x, y ∈ U, (iii) [d(x)2, d(y)2] − [x, y] ∈ Z(R) for all x, y ∈ U, (iv) [d(x2), d(y2)] − [x, y] ∈ Z(R) for all x, y ∈ U, (v) [d(x), d(y)] − [x2, y2] ∈ Z(R) for all x, y ∈ U. (2) a non-zero derivation d satisfying one of the following conditions: (i) d([d(x), d(y)]) − [x, y] ∈ Z(R) for all x, y ∈ U, (ii) d([d(x), d(y)]) + [x, y] ∈ Z(R) for all x, y ∈ U. Then R contains a non-zero central ideal.

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Commutativity with Derivations of Semiprime Rings

Let R be a ring, not necessarily with 1. Call potent (strongly potent) if for some natural (even natural number) n>1. The well known Jacobson's Commutavity Theorem [3] states : If every element x of a ring is potent, then R is commutative. Herstein [2, Theorem 3.13, pages 74] improved this: If, for every x, y in R, xy - yx is potent then R is commutative. Recently Machale [4] showed : If, for every x, y in R, xy+yx is strongly potent, then R is anticommutative. Yen [5] showed : If, for every x, y in R, either xy-yx is potent or xy+yx is strongly potent, then R is either commutative or anticommutative. Yen [6], also see [1], showed : If R is a semiprime ring and, for every x, y in R, xy+yx is potent, then R is commutative. The first author [1] showed : If R has 1 and, for every x, y in R, xy+yx is potent, then R is commutative. He also gave an example to show that in general if, for every x, y in R, xy+yx is potent, then R need not be commutative. Yen [7], proved : If R is a semiprime ring such that for every x, y in R, either xy+yx or xy-yx is potent then R is commutative. He conjectured in this paper : If R is a ring with 1 such that for every x, y in R either xy-yx is potent or xy+yx is potent, then R is commutative. Here we prove this conjecture. He also asked the following stronger question in a personal communication : Can we replace 'R with l' by $'R^2=R'$ in the above conjecture? Here we answer this question also as an application of the above result.

Ram Niwas Gupta

Papers

A Commutativity Theorem