Beck modules and alternative algebras

We set out the general theory of "Beck modules" in a variety of binary algebras and describe them as modules over suitable "universal enveloping" unital associative algebras. We develop a theory of "noncommutative partial differentiation" to pass from the equations of the variety to relations in a universal enveloping algebra. We pay particular attention to the case of alternative algebras, defined by a restricted associative law, and determine the Poincar & eacute; polynomial of the universal enveloping algebra in the abelian case.