Communications of the
Korean Mathematical Society CKMS

Let $R$ be a ring and $n$ a fixed non-negative integer. $\mathcal {TI}_n$ (resp. $\mathcal {TF}_n$) denotes the class of all right $R$-modules of $FGT$-injective dimensions at most $n$ (resp. all left $R$-modules of $FGT$-flat dimensions at most $n$). We prove that, if $R$ is a right $\Pi$-coherent ring, then every right $R$-module has a $\mathcal {TI}_n$-cover and every left $R$-module has a $\mathcal {TF}_n$-preenvelope. A right $R$-module $M$ is called $n$-$TI$-injective in case ${\rm Ext}^1(N,M)=0$ for any $N\in\mathcal{TI}_n$. A left $R$-module $F$ is said to be $n$-$TI$-flat if ${\rm Tor}_1(N,F)=0$ for any $N\in\mathcal{TI}_n$. Some properties of $n$-$TI$-injective and $n$-$TI$-flat modules and their relations with $\mathcal {TI}_n$-(pre)covers and $\mathcal {TF}_n$-preenvelopes are also studied.

Keywords: $\mathcal {TI}_n$-(pre)cover, $\mathcal {TF}_n$-preenvelope, $n$-$TI$-injective module, $n$-$TI$-flat module, weakly $n$-Gorenstein ring

MSC numbers: 16E10, 16E05, 16D80