Commutativity of near-rings with derivations by using algebraic substructures. (Q691671)

Let \(N\) denote a left near-ring, and \(d\) a nonzero derivation on \(N\). Define a nonempty subset \(U\) to be a semigroup left (resp. right) ideal if \(NU\subseteq U\) (resp. \(UN\subseteq U\)), and call \(U\) a semigroup ideal if it is both a semigroup left ideal and a semigroup right ideal. The paper presents additive and multiplicative commutativity theorems involving conditions satisfied by \(d\) on one or two subsets of \(N\). For example, it is shown that \(N\) must be a commutative ring if one of the following holds: (i) \(U\) is a semigroup right ideal containing \(a,b\) such that \(b\) and \(d(a)\) are not left zero divisors in \(N\), and \(d(uv)=d(vu)\) for all \(u,v\in U\); (ii) \(N\) is 3-prime, \(U\) is a nonzero semigroup right or left ideal, \(V\) is a nonzero semigroup ideal such that \(d^2(V)\neq\{0\}\), and \(g\) is a nonzero derivation on \(N\) such that \(d(v)g(u)=g(u)d(v)\) for all \(u\in U\) and \(v\in V\).