COMPLETELY PRIME SUBMODULES

Year 2013, Volume: 13 Issue: 13, 1 - 14, 01.06.2013

Nico J. Groenewald David Ssevviiri

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, βco(N). We call βco(M) = co.√0 the completely prime radical
of M. If R is a commutative ring, βco(M) = β(M) where β(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 βco(RR) and the ring
radical βco(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

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• N. J. Groenewald, D. Ssevviiri Department of Mathematics and Applied Mathematics Nelson Mandela Metropolitan University Port Elizabeth South Africa e-mails: [e-mail] (N. J. Groenewald) [e-mail] (D. Ssevviiri)