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This article introduces the concept of  S -semiprime submodules which are a generalization of semiprime submodules and  S -prime submodules. Let  M be a nonzero unital R-module, where  R is a commutative ring with a nonzero identity. Suppose that  S is a multiplicatively closed subset of  R . A submodule  P of  M is said to be an  S -semiprime submodule if there exists a fixed  s ∈ S , and whenever  r n m ∈ P for some  r ∈ R , m ∈ M , and  n ∈ ℕ , then  srm ∈ P . Also,  M is said to be an  S -reduced module if there exists (fixed)  s ∈ S , and whenever  r n m = 0 for some  r ∈ R , m ∈ M , and  n ∈ ℕ , then  srm = 0 . In addition, to give many examples and characterizations of  S -semiprime submodules and  S -reduced modules, we characterize a certain class of semiprime submodules and reduced modules in terms of these concepts.

S-Semiprime Submodules and S -Reduced Modules

S-Semiprime Submodules and $S$ -Reduced Modules