Geodesic
On identities of vector spaces embedded in finite associative algebras
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 3, pp. 69-77

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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.
Mots-clés : multiplicative vector space
Keywords: identity of vector space, $L$-variety, basis of identities, nonfinitely based space, nonfinitely based algebra. 
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     author = {I. M. Isaev and A. V. Kislitsin},
     title = {On identities of vector spaces embedded in finite associative algebras},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {69--77},
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     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a6/}
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I. M. Isaev; A. V. Kislitsin. On identities of vector spaces embedded in finite associative algebras. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 3, pp. 69-77. http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a6/