On the $S$-Krull dimension of a commutative ring

We explore the concept of the $S$-Krull dimension in commutative rings, extending classical notions of Krull dimension by incorporating multiplicative subsets $S$ of a ring $R$. We provide characterizations of $S$-prime, $S$-maximal, and $S$-minimal $S$-prime ideals, establishing foundational results crucial for understanding the $S$-Krull dimension. The paper also delves into properties of $S$-principal ideal $S$-domains, $S$-Artinian rings, and u-$S$-von Neumann regular rings, introducing an $S$-variant of the Division Algorithm. In the final section, the concept of $S$-height for $S$-prime ideals is defined, and the $S$-Krull dimension is analyzed through various characterizations and examples, with a particular focus on polynomial rings. We conclude with insights into the $S$-height of larger $S$-prime ideals of $R[X]$ and their intersections with $R$, contributing to a deeper understanding of the $S$-Krull dimension.