Pre-Lie Algebras, Their Multiplicative Lattice, and Idempotent Endomorphisms

We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring k with identity. These notions are applied to the study of pre-Lie k-algebras and, more generally, Lie-admissible k-algebras. Associating with any algebra \((A,\cdot )\) its sub-adjacent anticommutative algebra \((A,[-,-])\) is a functor from the category of k-algebras with pre-morphisms to the category of anticommutative k-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This allows to make use of all the notions concerning multiplicative lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra. We study idempotent endomorphisms of a pre-Lie algebra L, i.e., semidirect-product decompositions of L and bimodules over L.