Identities in Vector Spaces Embedded in Finite Associative Algebras

Abstract

We study identities in vector spaces embedded in finite associative linear algebras. We prove that the L-variety generated by the space of second order matrices over a finite field possesses finitely many L-subvarieties. We construct examples of a finite two-dimensional vector space, a finite four-dimensional linear algebra, and a ring consisting of 16 elements that have no finite basis of identities.