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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.

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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