Multiplicative bases and commutative semiartinian von Neumann regular algebras

TL;DR

This paper characterizes semiartinian von Neumann regular rings with primitive factors artinian, especially commutative $K$-algebras of countable type, by their dimension sequences, and constructs modules with specific injectivity properties over these algebras.

Contribution

It provides a constructive classification of certain commutative semiartinian regular $K$-algebras using dimension sequences and introduces modules with strict $ ext{lambda}$-injectivity over these algebras.

Findings

Dimension sequence $\\mathcal{D}_R$ classifies commutative semiartinian regular $K$-algebras of countable type.

Constructs unique $K$-algebras $B_{\alpha,n}$ from given dimension sequences.

Creates examples of strictly $\lambda$-injective modules over these algebras.

Abstract

Let $R$ be a semiartinian (von Neumann) regular ring with primitive factors artinian. The dimension sequence $\mathcal D _R$ is an invariant that captures the various skew-fields and dimensions occurring in the layers of the socle sequence of $R$. Though $\mathcal D _R$ does not determine $R$ up to an isomorphism even for rings of Loewy length $2$, we prove that it does so when $R$ is a commutative semiartinian regular $K$-algebra of countable type over a field $K$. The proof is constructive: given the sequence $\mathcal D$, we construct the unique $K$-algebra of countable type $R = B_{\alpha,n}$ such that $\mathcal D = \mathcal D _R$ by a transfinite iterative construction from the base case of the $K$-algebra $R(\aleph_0,K)$ consisting of all eventually constant sequences in $K^{\aleph_0}$. Moreover, we prove that the $K$-algebras $B_{\alpha,n}$ possess conormed strong multiplicative bases despite the fact that the ambient $K$-algebras $K^{\kappa}$ do not even have any bounded bases for any infinite cardinal $\kappa$. Recently, a study of the number of limit models in AECs of modules [1] has raised interest in the question of existence of strictly $\lambda$-injective modules for arbitrary infinite cardinals $\lambda$. In the final section, we construct examples of such modules over the $K$-algebra $R(\kappa,K)$ for each cardinal $\kappa \geq \lambda$. [1] M. Mazari-Armida, On limit models and parametrized noetherian rings, J. Algebra 669(2025), 58--74.