This article is cited in 5 papers

Mathematics

On almost nilpotent varieties in the class of commutative metabelian algebras

S. P. Mischenko^a, O. V. Shulezhko<sup>b

a</sup> Ulyanovsk State University, 42, Lev Tolstoy Street, Ulyanovsk, 432017, Russian Federation
^b Ulyanovsk State Pedagogical University named after I. N. Ulyanov, 4, Square of 100 from the date of birth of V. I. Lenin, Ulyanovsk, 432700, Russian Federation

Abstract: A well founded way of researching the linear algebra is the study of it using the identities, consequences of which is the identity of nilpotent. We know the Nagata-Higman's theorem that says that associative algebra with nil condition of limited index over a field of zero characteristic is nilpotent. It is well known the result of E. I. Zel'manov about nilpotent algebra with Engel identity. A set of linear algebras where a fixed set of identities takes place, following A. I. Maltsev, is called a variety. The variety is called almost nilpotent if it is not nilpotent, but each its own subvariety is nilpotent. Here in the case of the main field with zero characteristic, we proved that for any positive integer m there exist commutative metabelian almost nilpotent variety of exponent is equal to $m$.

Keywords: linear algebra, variety of algebras, almost nilpotent variety.

UDC: 512.554

Received: 11.03.2015

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