Gamma-derivations in prime and semiprime gamma-near-rings
Let \(M\) be a (right) \(\Gamma\)-near ring. A \(\Gamma\)-derivation of \(M\) is an additive endomorphism \(d\) of \(M\) satisfying \(d(x\gamma y)=x\gamma d(y)+d(x)\gamma y\) for all \(x,y\in M\) and \(\gamma\in\Gamma\). A \(\Gamma\)-derivation is said to act as a \(\Gamma\)-homomorphism on a subset \(S\) of \(M\) if \(d(x\gamma y)=d(x)\gamma d(y)\) for all \(x,y\in S\). It is shown that if \(M\) is semiprime, \(A\) is a left invariant subset of \(M\) containing \(0\), and \(\Gamma\)-derivation \(d\) acts as a \(\Gamma\)-homomorphism on \(A\), then \(d(A)=0\). Moreover, if \(M\) is a prime \(\Gamma\)-near-ring, \(A\) is a left invariant subset of \(M\) containing \(0\) and \(d\) acts as a \(\Gamma\)-homomorphism on \(A\), then \(d=0\).
scientific article; zbMATH DE number 1874245