Joseph Mastromatteo

Research interests: mathematics, group theory, module theory, and ring theory

My research is in the area of ring and module theory, a branch of abstract algebra. My focus has been the study of relative subprojectivity and relative pure subinjectivity, which provide ways to measure the projectivity and pure injectivity of a module, respectively.

Introduction

My research is a natural extension of projective, injective, and subinjective profiles, which are studied in the papers

• Poor Modules, The opposite of injectivity, A.N. Alahmadi, M. Alkan, S. R. Lopez-Permouth, Glasgow Math J., 2010, (journals.cambridge.org).
• An Alternative Perspective on Injectivity of Modules, P. Aydogdu, S.R. Lopez-Permouth, J. Algebra, 2011, (sciencedirect.com).
• Rings Whose Modules have Maximal or Minimal Injectivity Domains, N. Er, S. R. Lopez-Permouth, N. Sokmez, J. Algebra, 2011, (sciencedirect.com).
• Rings Whose Modules have Maximal or Minimal Projectivity Domains, C. Holston, S. R. Lopez-Permouth, N. Orhan Ertas, J. Pure Appl. Algebra, 2012, (sciencedirect.com).
• Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, S.R. Lopez-Permouth, J.E. Simental, J. Algebra, 2012, (arxiv.org).

Relative Subprojectivity and Subprojectivity Domains

Given a ring R and modules M and N, M is said to be N-subprojective if for every homomorphism f: MN and for every epimorphism g: KN, there exists a homomorphism h: MK such that gh=f. In other words, f can be factored as the composition of g with h. This notion is depicted in figure 1 below.

The subprojectivity domain of the module M is the collection of all modules N such that M is N-subprojective. M is called projective if it is N-subprojective relative to every module N, i.e. the subprojectivity domain of M is as large as possible, consisting of all modules. At this extreme, there is no distinction between the roles played by the subprojectivity domain and the projectivity domain. Interesting things happen, though, when we consider the subprojectivity domain of modules that are not projective.

Publications

1. J. Mastromatteo, C. Holston, S. R. Lopez-Permouth, and J. Simental, 2013, An alternative perspective on projectivity of modules, To appear in Glasgow J., (arxiv.org/abs/1206-5556)