On a category of extensions whose endomorphism rings have at most four maximal ideals

TL;DR

The paper investigates the category ${\mathcal{E}}$ of extensions $0 \to A_R \to B_R \to C_R \to 0$ where $A_R$ and $C_R$ are uniserial, showing the endomorphism ring $E_B$ of an object has at most four maximal ideals. It introduces four invariants, generalizing monogeny and epigeny classes, to classify finite direct sums in ${\mathcal{E}}$, and provides an explicit description of $E_B$ along with a complete isomorphism criterion for sums of extensions in the uniserial case. The main results establish how splitting of sequences relates to these invariants and prove a cancellation-type property for direct sums, supported by illustrative examples. The work connects semilocal endomorphism rings, completely prime ideals in the extension category, and a combinatorial isomorphism framework, advancing understanding of extensions with uniserial endpoints.

Abstract

We describe the endomorphism ring of a short exact sequences $0 \to A_R \to B_R \to C_R \to 0$ with $A_R$ and $C_R$ uniserial modules and the behavior of these short exact sequences as far as their direct sums are concerned.

Theorems & Definitions (48)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Remark 3.1
• Theorem 3.2
• proof
• Remark 3.3
• Remark 4.1