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# Algebra Seminar

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An alternative perspective on projectivity of modules, by Joe Mastromatteo (Ohio University)

What Seminar Nov 22, 2013 from 04:10 PM to 05:05 PM Morton 127 Sergio R. Lopez-Permouth 740-593-1258 vCal iCal

Abstract: 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 and is said to be N-subprojective if for every epimorphism $$g:B \rightarrow N$$ and homomorphism $$f:M \rightarrow N$$, there exists a homomorphism $$h:M \rightarrow B$$ 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 semilattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is 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. (This talk is based on joint work of Holston, Lopez-Permouth, Mastromatteo and Simental-Rodriguez.)

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