Openly semiprimitive projective module

Abstract

In this paper, a left module over an associative ring with identity is de¯ned to be openly semiprimitive (strongly semi- primitive, respectively) by the zero intersection of all maximal open fully invariant submodules (all maximal open submodules which are fully invariant, respectively) of it. For any projective module, the openly semiprimitivity of the projective module is an equivalent condition of the semiprimitivity of endomorphism ring of the pro- jective module and the strongly semiprimitivity of the projective module is an equivalent condition of the endomorphism ring of the projective module being a subdirect product of a set of subdivisions of division rings.