Multiplicative bases and commutative semiartinian von Neumann regular algebras

TL;DR

The paper develops dimension sequences $\mathcal{D}_R$ as a robust invariant for regular semiartinian rings with artinian primitive factors and proves that, in the commutative countable-type setting over a field $K$, $\mathcal{D}_R$ determines the ring up to a $K$-algebra isomorphism. It provides a constructive transfinite procedure to realize a unique $K$-algebra $R=B_{\alpha,n}$ from a given $\mathcal{D}$, starting from the base $R(\aleph_0,K)$ and building larger Loewy lengths via a ring-theoretic analogue of EGT. A key result is that for algebras of countable type, factor equivalence coincides with isomorphism, and these algebras admit conormed strong multiplicative bases, even though ambient $K$-algebras like $K^\kappa$ need not have bounded bases. The authors also construct strictly $\lambda$-injective modules over $R(\kappa,K)$ for every infinite cardinals $\kappa \ge \lambda$, resolving Eklof’s question in the affirmative within this framework and linking the algebraic structure to model-theoretic phenomena in AECs of modules.

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 $R = B_{α,n}$ 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 $B_{α,n}$ possess conormed strong multiplicative bases despite the fact that the ambient $K$-algebras $K^κ$ do not even have any bounded bases for any infinite cardinal $κ$. Recently, a study of the number of limit models in AECs of modules [1] has raised interest in the question of existence of strictly $λ$-injective modules for arbitrary infinite cardinals $λ$. In the final section, we construct examples of such modules over the $K$-algebra $R(κ,K)$ for each cardinal $κ\geq λ$. [1] M. Mazari-Armida, On limit models and parametrized noetherian rings, J. Algebra 669(2025), 58--74.

Theorems & Definitions (68)

• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Corollary 3.6
• Definition 3.7
• Definition 3.8
• Lemma 3.9