Countable separation property for associative algebras

TL;DR

This paper proves the countable separation property for certain associative algebras, including free associative algebras over any field, extending previous results beyond Noetherian cases.

Contribution

It establishes the CSP for associative algebras with simple modules and verifies it for free associative algebras over arbitrary fields, broadening the scope of prior work.

Findings

CSP holds for associative algebras with simple modules and trivial endomorphisms.

CSP verified for free associative algebras over any field.

Extends CSP results beyond Noetherian algebras.

Abstract

For an associative algebra $A$ with a simple module $M$ with trivial endomorphisms and trivial annihilator we verify the countable separation property (CSP), i.e. we prove that there exists a list of nonzero elements $a_1, a_2,\ldots$ of $A$ such that every two-sided ideal of $A$ contains at least one such $a_i$. Based on this result we verify the countable separation property for a free associative algebra with finite or countable set of generators over any field. The countable separation property was studied before in the works of Dixmier and others but only in the context of Noetherian algebras (and a free associative algebra is very far from being Noetherian).