  We study the relationship of backcrossing algebras with mutation algebras and
algebras satisfying $\omega$-polynomial identities: we show that in a
backcrossing algebra every element of weight 1 generates a mutation algebra and
that for any polynomial identity $f$ there is a backcrossing algebra satisfying
$f$. We give a criterion for the existence of idempotent in the case of baric
algebras satisfying a nonhomogeneous polynomial identity and containing a
backcrossing subalgebra. We give numerous genetic interpretations of the
algebraic results.

Commutative Algebra