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Affine representability and decision procedures for commutativity theorems for rings and algebras

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Abstract

We consider applications of a finitary version of the Affine Representability theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne. Using this result we are able to show that when given a finite set of polynomial identities, there is an algorithm that terminates after a finite number of steps which decides whether these identities force a ring to be commutative. We then revisit old commutativity theorems of Jacobson and Herstein in light of this algorithm and obtain general results in this vein. In addition, we completely characterize the homogeneous multilinear identities that imply the commutativity of a ring.

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Funding

The work of Jason P. Bell was supported by NSERC Discovery Grant RGPIN-2016-03632. The work of Peter V. Danchev was partially supported by the Bulgarian National Science Fund under Grant KP-06 No 32/1 of December 07, 2019.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jason P. Bell

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” str., bl. 8, 1113, Sofia, Bulgaria

    Peter V. Danchev

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  1. Jason P. Bell
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  2. Peter V. Danchev
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Correspondence to Jason P. Bell.

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Bell, J.P., Danchev, P.V. Affine representability and decision procedures for commutativity theorems for rings and algebras. Isr. J. Math. 249, 121–166 (2022). https://doi.org/10.1007/s11856-022-2309-3

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  • DOI: https://doi.org/10.1007/s11856-022-2309-3

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