$n$-cotorsion pairs over formal triangular matrix rings

TL;DR

This paper investigates special classes of modules over formal triangular matrix rings and constructs $n$-cotorsion pairs over such rings from those over component rings, providing concrete examples.

Contribution

It introduces a method to build $n$-cotorsion pairs over triangular matrix rings from pairs over individual rings, expanding the understanding of module classes in this context.

Findings

Constructed $n$-cotorsion pairs over triangular matrix rings from pairs over component rings.

Provided explicit examples illustrating the main theoretical results.

Extended the theory of cotorsion pairs to formal triangular matrix rings.

Abstract

Let $\Lambda=\begin{pmatrix}A & 0 \\U & B \end{pmatrix}$ 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 $\Lambda$. Further, using these special classes, we construct a left (resp. right) $n$-cotorsion pair over the formal triangular matrix ring $\Lambda$ from left (resp. right) $n$-cotorsion pairs over $A$ and $B$. Finally, we give an example to illustrate our main result.