Simple derivations on tensor product of polynomial algebras

Let A be an unique factorization domain containing a field k of characteristic zero and let A[X] and A[Y ] be two k-algebras. Let d1 and d2 be two generalized triangular k-derivations of A[X] and A[Y ], respectively. Denote the unique k-derivation d1 ⊗ 1 + 1 ⊗ d2 of A[X,Y ] by d1 ⊕ d2. Then with some conditions on d1 and d2, it is shown that d1 ⊕ d2 is a simple derivation of A[X,Y ] if and only if A[X] is d1-simple and A[Y ] is d2-simple. We also show that if d1 and d2 are positively homogeneous derivations and d2 is a generalized triangular derivation, then d1 ⊕ d2 is simple derivation of A[X,Y ] if and only if d1 is a simple derivation of A[X] and d2 is a simple derivation of A[Y ].