Identities of sum of two PI-algebras in the case of positive characteristic

TL;DR

This paper investigates whether the sum of two PI subrings in an associative ring over a field of positive characteristic also satisfies a polynomial identity, providing new conditions and bounds for such identities.

Contribution

It establishes new conditions under which the sum of two PI subrings in positive characteristic satisfies a polynomial identity, extending previous results.

Findings

Derived upper bounds on degrees of identities

Established lower bounds on degrees of identities

Provided conditions guaranteeing polynomial identities in sums of PI subrings

Abstract

We consider the following question posted by K.I. Beidar and A.V. Mikhalev in 1995 for an associative ring $R=R_1+R_2$: is it true that if the subrings $R_1$ and $R_2$ satisfy polynomial identities, then $R$ also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on $R_1$ and $R_2$ that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of $R$.