COMPLETELY PRIME SUBMODULES

We generalize completely prime ideals in rings to submodules in modules. The notion of multiplicative systems of rings is generalized to modules. Let N be a submodule of a left R-module M. Define co.root N := {m is an element of M : every multiplicative system containing m meets N}. It is shown that co.root N is equal to the intersection of all completely prime submodules of M containing N, beta(co) (N). We call beta(co) (M) = co.root 0 the completely prime radical of M. If R is a commutative ring, beta(co) (M) = beta(M) where beta(M) denotes the prime radical of M. beta(co) is a complete Hoehnke radical which is neither hereditary nor idempotent and hence not a Kurosh-Amistur radical. The torsion theory induced by beta(co) is discussed. The module radical beta(co) (R-R) and the ring radical beta(co) (R) are compared. We show that the class of all completely prime modules, R-M for which RM not equal 0 is special.