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.√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: nico.groenewald@nmmu.ac.za (N. J. Groenewald) david.ssevviiri@nmmu.ac.za (D. Ssevviiri)
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI