Countable separation property for associative 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 $a_1,a_2,\dots$ 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).