Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $u$-$S$-projective dimension and $u$-$S$-injective dimension of an $R$-module, and then explore the $u$-$S$-global dimension $u$-$S$-\gld$(R)$ of a commutative ring $R$, i.e., the supremum of $u$-$S$-projective dimensions of all $R$-modules. Finally, we investigate $u$-$S$-global dimensions of factor rings and polynomial rings.

Keywords: $u$-$S$-projective dimension, $u$-$S$-injective dimension, $u$-$S$-global dimension, factor ring, polynomial ring

MSC numbers: 13D05, 13D07

Supported by: The first author was supported by National Natural Science Foundation of China (No. 12201361).