Idempotent pre-endomorphisms of algebras

In the study of pre-Lie algebras, concept pre-morphism arises naturally as a generalization standard notion morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring k with identity, and dualized in various ways to generalized morphisms (related pre-Jordan algebras) anti-pre-morphisms anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent are related semidirect-product decompositions sub-adjacent anticommutative algebra.