Semi r-ideals of commutative rings

For commutative rings with identity, we introduce and study the concept of semi -ideals which is a kind of generalization of both -ideals and semiprime ideals. A proper ideal of a commutative ring is called semi -ideal if whenever ∈ and ) = 0, then ∈ . Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi -ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi -ideals of rings to semi -submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the -annihilator condition and justify when they are semi -submodules.