A study of <i>S</i> -maximal ideals and their structural applications

This paper investigates S-maximal ideals in commutative rings, where S is multiplicatively closed. We examine their core properties, localization behavior, and relation to S-comaximality. An S-analogue of the Chinese Remainder Theorem is established, along with a characterization of the S-Jacobson radical via S-invertibility. Applications include S-quasi-local rings, a Nakayama-type lemma for S-finite ideals, and a module-theoretic extension of the S-Chinese Remainder Theorem. Examples clarify how S-theoretic concepts differ from classical ones, offering new tools for analyzing ring and module structures.