  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 $L$.
