Commutativity theorems for rings with differential identities on Jordan ideals. (English) Zbl 1299.16018

The following situation is investigated: Let \(R\) be an associative ring with center \(Z(R)\) and denote \([x,y]=xy-yx\), \(x\circ y=xy+yx\) for all \(x,y\in R\). Let \(x\mapsto x^*\) be an involution on \(R\) (i.e., an additive mapping such that \((xy)^*=y^*x^*\) and \((x^*)^*=x\) for all \(x,y\in R\)). Suppose further that \(R\) is 2-torsionfree \(*\)-prime (i.e., \(2x=0\) implies \(x=0\) and \(aRb=aRb^*=0\) implies that either \(a=0\) or \(b=0\)) and \(J\) is a non-zero \(*\)-Jordan ideal of \(R\) (i.e., \(J\) is an additive subgroup of \(R\) with \(J^*=J\) and \(u\circ r\in J\) for all \(u\in J\) and \(r\in R\)).
The main results of the paper show that \(R\) is commutative whenever it admits a non-zero derivation \(d\) such that one of the following four conditions is satisfied: (i) \(d([x,y])\in Z(R)\) for all \(x,y\in J\) (Theorem 1); (ii) for all \(x,y\in J\), either \(d([x,y])-[x,y]\in Z(R)\) or \(d([x,y])+[x,y]\in Z(R)\) (Theorem 4); (iii) \(d(x\circ y)\in Z(R)\) for all \(x,y\in J\) (Theorem 6); (iv) for all \(x,y\in J\), either \(d(x\circ y)-x\circ y\in Z(R)\) or \(d(x\circ y)+x\circ y\in Z(R)\) (Theorem 9).


16R50 Other kinds of identities (generalized polynomial, rational, involution)
16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
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