Prime Submodules and Local Gabriel Correspondence in σ[M]

Abstract

We consider the concept of prime submodule defined by Raggi et al. [7]. We find equivalent conditions for a module M progenerator in σ[M], with τ _M -Gabriel dimension, to have a one-to-one correspondence between the set of isomorphism classes of indecomposable τ-torsion free injective modules in σ[M] and the set of τ-pure submodules prime in M, where τ is a hereditary torsion theory in σ[M]. Also we give a relation between the concept of prime M-ideal given by Beachy and the concept of prime submodule in M. We obtain that if M is progenerator in σ[M], then these concepts are equivalent.

Key Words:

• Gabriel correspondence
• Preradicals
• Prime submodules
• Torsion theory

2000 Mathematics Subject Classification:

• Primary 16S90
• Secondary 16D50, 16P50, 16P70

Acknowledgments

Dedicated to the memory of Lucía Pérez Maya (1938–2007) and Bertha Montes Flores (1927–2010).

Notes

Communicated by T. Albu.