Unification modulo ACUI plus distributivity axioms

E-unification problems are central in automated deduction. In this work, we consider unification modulo theories that extend the well-known ACI or ACUI by adding a binary symbol "*" that distributes over the AC(U) I-symbol "+." If this distributivity is one-sided (say, to the left), we get the theory denoted AC(U)ID_l; we show that AC(U)ID _l-unification is DEXPTIME-complete. If "*" is assumed two-sided distributive over "+," we get the theory denoted AC(U)ID; we show unification modulo AC(U)ID to be NEXPTIME-decidable and DEXPTIME-hard. Both AC(U)ID_l and AC(U)ID seem to be of practical interest, for example, in the analysis of programs modeled in terms of process algebras. Our results, for the two theories considered, are obtained through two entirely different lines of reasoning. A consequence of our methods of proof is that, modulo the theory that adds to AC(U)ID the assumption that "*" is associative-commutative, or just associative, unification is undecidable.