# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2021; 58(6): 1483-1494

Published online November 30, 2021 https://doi.org/10.4134/BKMS.b201049

## Gorenstein flat-cotorsion modules over formal triangular matrix rings

Dejun Wu

Lanzhou University of Technology

### Abstract

Let $A$ and $B$ be rings and $U$ be a $(B,A)$-bimodule. If $_BU$ has finite flat dimension, $U_A$ has finite flat dimension and $\tpa{U}{C}$ is a cotorsion left $B$-module for any cotorsion left $A$-module $C$, then the Gorenstein flat-cotorsion modules over the formal triangular matrix ring $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\ \end{smallmatrix}\right)$ are explicitly described. As an application, it is proven that each Gorenstein flat-cotorsion left $T$-module is flat-cotorsion if and only if every Gorenstein flat-cotorsion left $A$-module and $B$-module is flat-cotorsion. In addition, Gorenstein flat-cotorsion dimensions over the formal triangular matrix ring $T$ are studied.

Keywords: Formal triangular matrix ring, Gorenstein flat-cotorsion module

MSC numbers: 16E10, 16G50, 18G25

Supported by: The author was partly supported by NSF of China grants 11761047 and 11861043.