On c-S-(weak) global dimensions

In 2002, Anderson and Dumitrescu introduced [Formula: see text]-Noetherian rings, a significant advancement in ring theory. Later, in 2018, Bennis and El Hajoui expanded on this by introducing two new classes of rings that generalize the concept of coherent rings: [Formula: see text]-coherent rings, where every finitely generated ideal is [Formula: see text]-finitely presented, and [Formula: see text]-[Formula: see text]-coherent rings, where every [Formula: see text]-finite ideal is [Formula: see text]-finitely presented. This paper builds on these concepts to introduce the [Formula: see text]-[Formula: see text]-(weak) global dimension for rings with a multiplicative subset [Formula: see text], offering a new perspective on ring structures. We also explore [Formula: see text]-[Formula: see text]-variants of specific ring types, including semisimple, von Neumann, and (semi)hereditary rings. Finally, we extend the generalization of [Formula: see text]-rings and [Formula: see text]-coherent rings, originally introduced by Costa in 1994, to rings with a multiplicative subset [Formula: see text], deepening the understanding of ring theory.