On semigroup ideals and generalized \(n\)-derivations in near-rings. (Q2786759)

Let \(N\) be a zero-symmetric near-ring with multiplicative center \(Z\), and define a semigroup ideal of \(N\) to be a nonempty subset \(U\) such that \(UN\subseteq U\) and \(NU\subseteq U\). An \(n\)-derivation is an \(n\)-additive map \(D\colon N\times N\times\cdots\times N\to N\), where there are \(n\) factors in the direct product, such that fixing every argument except the \(i\)-th yields a derivation \(D_i\) on \(N\); and a generalized \(n\)-derivation with associated \(n\)-derivation \(D\) is an \(n\)-additive map \(F\colon N\times N\times\cdots\times N\to N\) such that fixing every argument except the \(i\)-th yields a generalized derivation \(F_i\) with associated derivation \(D_i\).NEWLINENEWLINE The first theorem asserts that if \(N\) is 3-prime and \(U_1,U_2,\ldots,U_n\) are nonzero semigroup ideals, and if \(N\) admits a generalized \(n\)-derivation \(F\) such that \(F(U_1,U_2,\ldots,U_n)\subseteq Z\), then \(N\) is a commutative ring. There are several other results asserting that certain constraints involving generalized \(n\)-derivations and semigroup ideals imply some additive or multiplicative commutativity.