On the Characterization and Properties of <i>S</i> -Jacobson radical Submodules

This paper introduces and studies the S-Jacobson radical submodule J S (M), a generalization of the classical Jacobson radical submodule of a module with respect to a multiplicatively closed subset S of a commutative ring R. We establish a fundamental characterization of J S (M) as the set of S-non-generators, examine its behavior under module homomorphisms, and investigate its relationships with other structural notions, including the S-radical Rad S (M) and the S-Jacobson radical J S (R) of the base ring. Our main results include natural inclusion theorems, sufficient conditions for equality among these radicals, and an S-version of Nakayamas lemma for S-finitely generated modules. Several examples are provided to illustrate the theory and to highlight essential differences from the classical Jacobson radical framework.