## 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).

### MSC:

 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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