Bounded Modules

Lei R be a commutative ring with identity, and let M be a unitary (left) Rmodule. The ideal annR M = {r є R;rm = 0 V mє M} plays a central role in our work. In fact, we shall be concerned with the ease wrhcre ann RM = ann R(x) for some X є M such modules will be called bounded modules.lt turns out that there are many classes of modules properly contained in the class of bounded modules such as cyclic modules, torsion-free modules,faithful multiplication modules, prime modules and cyclic modules over their endomorphism rings. Also, using boundedness of modules, we showed that: - The classes of injective modules modulo annihilator and quasi-injective modules arc equivalent. - The classes of faithful modules and compactly faithful modules are equivalent.