Countable separation property for associative algebras

TL;DR

Addressing CSP for associative algebras, the paper shows that if $A$ is countable-dimensional with 1 over $\mathbb K$ and admits a simple $A$-module $M$ with trivial annihilator and ${\rm Hom}_A(M,M)=\mathbb K$, then $A$ is E-CSP; the same holds for the base-change algebra $A\otimes_{\mathbb K}\widetilde{\mathbb K}$. This enables CSP for free associative algebras $F_n(\mathbb K)$ (finite generators) and $F_{\mathbb N}(\mathbb K)$ (countably generated) by constructing explicit separating lists and exploiting countable exhaustions. The results extend earlier CSP work by Dixmier, which focused on Noetherian cases, and reveal a structural way to decompose two-sided ideals into countably many families via the separating elements; the framework also ties CSP to primitive algebras, center considerations, and stabilization concepts. Overall, the findings provide a practical criterion to guarantee CSP/E-CSP in broad classes of associative algebras, including a direct route to CSP for key non-Noetherian examples like free 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).

Theorems & Definitions (15)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Remark 1
• Corollary 1
• Theorem 3
• proof : Proof of Theorem \ref{['T:stab']}(a)
• Lemma 1
• proof