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On matrix algebras with two generators and on embedding of PI-algebras. (English. Russian original) Zbl 0787.17003
Russ. Math. Surv. 47, No. 4, 216-217 (1992); translation from Usp. Mat. Nauk 47, No. 4(286), 199-200 (1992).
Let $$R$$ be any algebra with $$k$$ generators, over a commutative- associative ring $$\Phi$$ with 1; and let $$R^ \#$$ be the algebra obtained from $$R$$ by adjoining an identity. Also, for $$x$$ a real number, let $$m(x)$$ denote the smallest integer $$m$$ such that $$m\geq x$$. Then for $$n\geq 2m(\sqrt{k})+1$$, the matrix algebra $$M_ n(R^ \#)$$ is generated by two elements.
This result has the following applications: (1) If $$R$$ is a finitely- generated associative PI-algebra, then $$R$$ can be embedded in an associative PI-algebra with two generators. (2) If $$R$$ is a finitely- generated special Jordan PI-algebra and $$1/2\in\Phi$$, then $$R$$ can be embedded in a special Jordan PI-algebra with two generators.

##### MSC:
 17A99 General nonassociative rings 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings 17C05 Identities and free Jordan structures
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