Rings with Polynomial Identity and Centrally Essential Rings

TL;DR

This paper constructs examples of centrally essential algebras over fields of prime characteristic that do not satisfy polynomial identities and are not algebraic over their centers, challenging existing assumptions.

Contribution

It demonstrates the existence of centrally essential algebras with specific properties that were previously unknown or unconfirmed.

Findings

Existence of centrally essential algebras not satisfying polynomial identities

Construction over fields of prime characteristic

Algebras are not algebraic over their centers

Abstract

It is proved that for any prime integer $p$ and each field $F$ of characteristic $p$, there exists a centrally essential $F$-algebra which is not a PI-ring and is not algebraic over its center. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.