On generalized \((\theta,\phi)\)-derivations in rings. (English) Zbl 1076.16512

Ashraf, Mohammad; Al-Shammakh, Wafa S. M.

Int. J. Math. Game Theory Algebra 12, No. 4, 295-300 (2002).

Summary: Let \(R,S\) be rings and let \(\theta,\phi\) be homomorphisms of \(S\) into \(R\). Suppose that \(M\) is an \(R\)-bimodule. An additive mapping \(F\colon S\to M\) is called a generalized \((\theta,\phi)\)-derivation (resp. generalized Jordan \((\theta,\phi)\)-derivation) if there exists a \((\theta,\phi)\)-derivation \(d\colon S\to M\) such that \(F(xy)=F(x)\theta(y)+\phi(x)d(y)\) (resp. \(F(x^2)=F(x)\theta(x)+\phi(x)d(x)\)) holds for all \(x,y\in S\). In the present paper, it is shown that if \(R\) is non-commutative, \(M\) is \(2\)-torsion free, \(\theta\) is onto, and if \(mRx=0\) with \(m\in M\), \(x\in R\) implies \(m=0\) or \(x=0\), then every generalized Jordan \((\theta,\phi)\)-derivation \(F\colon S\to M\) is a generalized \((\theta,\phi)\)-derivation. Further, a related result for an arbitrary ring \(R\) is also obtained.

MSC:

16W25	Derivations, actions of Lie algebras
16N60	Prime and semiprime associative rings

Keywords:

prime rings; generalized Jordan derivations; inner derivations; torsionfree rings; generalized derivations