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AN ALTERNATIVE PERSPECTIVE ON PROJECTIVITY OF MODULES

Published online by Cambridge University Press:  26 August 2014

CHRIS HOLSTON ,
SERGIO R. LÓPEZ-PERMOUTH ,
JOSEPH MASTROMATTEO  and
JOSÉ E. SIMENTAL-RODRÍGUEZ
CHRIS HOLSTON
Affiliation:
Department of Mathematics. Ohio University, Athens, OH 45701, USA e-mails: holston@ohio.edu, lopez@ohio.edu, jm424809@ohio.edu, js224910@ohio.edu
SERGIO R. LÓPEZ-PERMOUTH
Affiliation:
Department of Mathematics. Ohio University, Athens, OH 45701, USA e-mails: holston@ohio.edu, lopez@ohio.edu, jm424809@ohio.edu, js224910@ohio.edu
JOSEPH MASTROMATTEO
Affiliation:
Department of Mathematics. Ohio University, Athens, OH 45701, USA e-mails: holston@ohio.edu, lopez@ohio.edu, jm424809@ohio.edu, js224910@ohio.edu
JOSÉ E. SIMENTAL-RODRÍGUEZ
Affiliation:
Department of Mathematics. Ohio University, Athens, OH 45701, USA e-mails: holston@ohio.edu, lopez@ohio.edu, jm424809@ohio.edu, js224910@ohio.edu
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Abstract

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We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M is said to be N-subprojective if for every epimorphism g : B → N and homomorphism f : M → N, there exists a homomorphism h : M → B such that gh = f. For a module M, the subprojectivity domain of M is defined to be the collection of all modules N such that M is N-subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semi-lattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is as smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (p-indigent), and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.

Keywords

16D4016D5016D80

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 83 - 99
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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