Simple derivations on tensor product of polynomial algebras

Abstract

Let [Formula: see text] be an unique factorization domain containing a field [Formula: see text] of characteristic zero and let [Formula: see text] and [Formula: see text] be two [Formula: see text]-algebras. Let [Formula: see text] and [Formula: see text] be two generalized triangular [Formula: see text]-derivations of [Formula: see text] and [Formula: see text], respectively. Denote the unique [Formula: see text]-derivation [Formula: see text] of [Formula: see text] by [Formula: see text]. Then with some conditions on [Formula: see text] and [Formula: see text], it is shown that [Formula: see text] is a simple derivation of [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-simple and [Formula: see text] is [Formula: see text]-simple. We also show that if [Formula: see text] and [Formula: see text] are positively homogeneous derivations and [Formula: see text] is a generalized triangular derivation, then [Formula: see text] is simple derivation of [Formula: see text] if and only if [Formula: see text] is a simple derivation of [Formula: see text] and [Formula: see text] is a simple derivation of [Formula: see text].