О многообразиях, порожденных кольцом, конечномерным над центроидом

What is it about?

It is known that varieties of associative algebras over an infinite field are locally generated by a finite-dimensional algebra [A. R. Kemer, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 726–753; MR1073084]. However, this is not true for algebras over a finite field. The simplest example is a variety generated by the semidirect product of a finite field $k$ and the polynomial ring $k[x]$. A more interesting example is a variety generated by matrices of the form $\left(\smallmatrix P&\ast\\ 0 &P^q\endsmallmatrix \right)$, where $P\in k[x]$ and the field $k$ satisfies the identity $z^q=z$. Nevertheless, any variety of associative algebras over an arbitrary associative-commutative ring $\Phi$ is generated by a subalgebra of an algebra that is finite-dimensional over the center [A. Ya. Belov, Ph.D. dissertation, Mosk. Gos. Univ., Moscow, 2002; per bibl.]. The main theorem of the paper is as follows. "Theorem 1. The ideal of the identities of a variety $\germ M$ of algebras of arbitrary signature, which is generated by an algebra that is finite-dimensional over the centroid, is the intersection of the ideals of the identities of a finite algebra and a homogeneous variety. If the variety $\germ M$ is unitarily closed, then it is generated by an algebra over the center.

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