Jordan ideals and \((\alpha, \beta)\)-derivations on 3-prime near-rings and rings (Q6545026)

A (left) near-ring is a triple \((N,+,.)\), where \((N,+)\) is a group (not necessarily abelian), \((N,.)\) is a semigroup, and the left distributive law \(a.(b+c)=a.b+a.c\) holds for all \(a,b,c \in N\). \(N\) is called prime if \(xNy=\{0\}\) implies \( x=0\) or \( y=0\). A right (resp. left) near-ring \( N\) is said to be zero symmetric if \(x0 = 0\), (resp. \(0x = 0\),) for all \(x \in N\) (note that right distributivity implies \(0x = 0\) and the left distributivity implies \(x0 = 0\)). An additive mapping \(d : N \to N\) is called an \((\alpha, \beta)\)-derivation if there exist \(\alpha, \beta : N \to N\) such that \(d(x y) = d(x)\alpha(y) + \beta(x)d(y)\) for all \( x, y \in N\) and an additive mapping \(f : N \to N\) is called a \((\alpha, \beta)\)-generalized derivation if there exist endomorphisms \(\alpha, \beta : N \to N\) and an \((\alpha, \beta)\)-derivation d such that \(f (x y) = f (x)\alpha(y) + \beta(x)d(y)\) for all \(x, y \in N\). An additive map \(G : N \to N\) is said to be a left \(\alpha\)-multiplier, if there exists a map \(\alpha : N \to N\) such that \(G(x y) = G(x)\alpha(y)\) for all \(x, y \in N\). In the article under review, the authors characterize the commutativity of \(3\)-prime near rings and obtain some results left \(\alpha\)-multipliers on left near rings and right near rings. This result is proved via direct calculation, much the same as in other papers with similar results.