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