The Variety of Near-Rings is Generated by Its Finite Members

Abstract

We show that the variety of near-rings and the variety of zero-symmetric near-rings are both generated by their finite members. We show this in a more general context: if a variety \({\cal V}\) is generated by a class of algebras \({\cal F}\), then the variety of \({\cal V}\)-composition algebras is generated by the class of all full function algebras on direct products of finitely many copies of algebras in \({\cal F}\).