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.