Primitivity of prime countable-dimensional regular algebras

TL;DR

This paper proves that certain countable-dimensional prime von Neumann regular algebras over a field are primitive, addressing a specific case of a question posed by Kaplansky.

Contribution

It establishes the primitivity of prime countable-dimensional regular algebras, advancing understanding in algebraic structure theory.

Findings

Prime countable-dimensional regular algebras over a field are primitive.

Answers a special case of Kaplansky's question.

Contributes to the classification of von Neumann regular algebras.

Abstract

Let $k$ be a field and let $R$ be a countable dimensional prime von Neumann regular $k$-algebra. We show that $R$ is primitive, answering a special case of a question of Kaplansky.