COMPLETELY PRIME SUBMODULES

Authors : Nico J GROENEWALD, David SSEVVİİRİ

Pages : 1-14

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Publication Date : 2013-06-01

Article Type : Research Paper

Abstract :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.√N := {m ∈ M : every multiplicative system containing m meets N}. It is shown that co.√N is equal to the intersection of all completely prime submodules of M containing N, βcoinsert ignore into journalissuearticles values(N);. We call βcoinsert ignore into journalissuearticles values(M); = co.√0 the completely prime radical of M. If R is a commutative ring, βcoinsert ignore into journalissuearticles values(M); = βinsert ignore into journalissuearticles values(M); where βinsert ignore into journalissuearticles values(M); denotes the prime radical of M. βco is a complete Hoehnke radical which is neither hereditary nor idempotent and hence not a Kurosh-Amistur radical. The torsion theory induced by βco is discussed. The module radical βcoinsert ignore into journalissuearticles values(RR); and the ring radical βcoinsert ignore into journalissuearticles values(R); are compared. We show that the class of all completely prime modules, RM for which RM 6= 0 is special.
Keywords : completely prime submodules, completely prime radical of a module, special class of modules and multiplicative system of modules