Commutativity theorems for *-prime rings with differential identities on Jordan ideals.
In a (*-) prime ring \(R\) the authors study special identities for Jordan ideals of \(R\) that involve a derivation of \(R\) and commutators or anti-commutators. Let \(R\) be a 2-torsion free ring with derivation \(d\) and nonzero Jordan ideal \(J\): \(J\) is a subgroup of \((R,+)\) so that for all \(x\in J\) and \(r\in R\), \(x\circ r=xr+rx\in J\). For \(x,y\in R\) set \([x,y]=xy-yx\). When \(R\) has an involution * then \(R\) is *-prime if for any \(a,b\in R\), \(aRb=0=aRb^*\) implies that \(a=0\) or \(b=0\). The main results of the authors assume that \(R\) is *-prime, \(d\) is not zero, \(d*=*d\) as functions on \(R\), and \(J=J^*\). Then \(R\) must be commutative if any of \([d(x),d(y)]=[x,y]\), \([d(x),d(y)]=0\), or \([d(x),d(y)]=d([x,y])\) is an identity on \(J\). Also, for any \(d\), if \(d(x)\circ d(y)=0\) or \(d(x)\circ d(y)=d(x\circ y)\) is an identity for \(J\), then \(d=0\). For any prime \(R\), \(d\), and \(J\) as initially defined above, each of these five identities yields the corresponding statement for \(R\).
scientific article; zbMATH DE number 6123892