On identities of vector spaces embedded in finite associative algebras

In this paper we study identities of vector spaces embedded in finite associative linear algebras. We prove that a $L$-variety generated by the space of matrices of second order over a finite field has a finite number of $L$-subvarieties. We constructed an example of a finite two-dimensional vector space which has no finite basis of identities. As a corollary, we constructed an example of a finite four-dimensional linear algebra without finite basis of identities. In particular, the authors constructed an example of a ring consisting of 16 elements which has no finite basis of identities.