On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings

This study introduces and investigates the idea of $S$-pm-rings, a generalization of pm-rings in the context of commutative rings with a multiplicatively closed subset $S$. We prove that a ring $R$ is an $S$-pm-ring if and only if its $S$-maximal spectrum is a retract (specifically, a deformation retract) of its $S$-prime spectrum. Furthermore, we establish the equivalence of the $S$-pm-ring property to the normality of the $S$-prime spectrum and the Hausdorff property of the $S$-maximal spectrum. We also explore the relationship between $S$-pm-rings and $S$-clean rings, demonstrating that every $S$-local ring is $S$-clean, and every $S$-clean ring is an $S$-pm-ring. These results extend classical theorems in commutative algebra and algebraic geometry to the $S$-version context.