$n$-cotorsion pairs over formal triangular matrix rings

TL;DR

The paper addresses how to extend $n$-cotorsion theory to formal triangular matrix rings by constructing left and right $n$-cotorsion pairs in $\\Lambda$-Mod from hereditary pairs on $A$-Mod and $B$-Mod. It introduces three special classes $\\mathfrak{A}^\\mathcal{C}_\\mathcal{E}$, $\\mathfrak{P}^\\mathcal{C}_\\mathcal{E}$, and $\\mathfrak{I}^\\mathcal{C}_\\mathcal{E}$ and proves that, under vanishing Tor $ (\\operatorname{Tor}^A_j(U,\\mathcal{C})=0$ for $1\\le j\\le n+1$) and Ext conditions $(\\operatorname{Ext}^j_B(U,\\mathcal{F})=0$ for $1\\le j\\le n+1$) plus extension-closure, the assembled pairs form hereditary $n$-cotorsion pairs in $\\Lambda$-Mod. The results yield explicit lifting criteria for right and left $n$-cotorsion structures and provide an illustrative example, contributing to a deeper understanding of cotorsion theory on triangular rings with potential applications in representation theory and module categories of composite rings.

Abstract

Let $Λ=\begin{pmatrix}A & 0 \\U & B \end{pmatrix}$ be a formal triangular matrix ring where $A,B$ are rings and $U$ is a $(B,A)$-bimodule. In this paper, we study some special classes over the formal triangular matrix ring $Λ$. Further, using these special classes, we construct a left (resp. right) $n$-cotorsion pair over the formal triangular matrix ring $Λ$ from left (resp. right) $n$-cotorsion pairs over $A$ and $B$. Finally, we give an example to illustrate our main result.

Theorems & Definitions (32)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Definition 2.5
• Theorem 2.6
• Proposition 2.7
• Lemma 2.8
• Definition 3.1