OUOSU ring theory seminar
Pureinjectivity from an alternative perspective, by Joseph Mastromatteo (Ohio University)
What 


When 
Apr 04, 2014 from 04:45 PM to 05:45 PM 
Where  CH 240 (OSU, Columbus) 
Contact Name  Sergio R. Lopez 
Contact Phone  740 593 1258 
Add event to calendar 
vCal iCal 
Abstract:
The study of injectivity has frequently been approached from the perspective of relative notions. For a module M, its injectivity domain consists of all modules N such that M is injective relative to N (or Ninjective). In [1], the authors view the class of all injectivity domains of modules over a ring R as an ordered structure (the injective profile of the ring R) and investigate the interactions between properties of that injective profile and those of the ring itself. In [2], the authors explore an alternative perspective: instead of using the injectivity domain of a module M as a mean to gauge the extent of its injectivity, they consider the socalled subinjectivity domain of M. M is Nsubinjective means that if for every extension K of N and every homomorphism f : N > M there exists a homomorphism g : K > M such that g_N=f. In [3], the pureinjectivity profile of a ring is introduced as an analog to the injectivity profile of [1], but instead consider only pure extensions K of N.
In this talk, we explore the notion of relative puresubinjectivity. A module M is said to be Npuresubinjective if every homomorphism from N to M can be extended to a homomorphism from K to M, where K is a pure extension of N. In particular, we give characterizations of right pure hereditary rings, von Neumann regular rings, and semisimple rings, by comparing the puresubinjectivity domains with the other domains of pureinjectivity and injectivity. We also consider when the puresubinjectivity domain of a module is closed under pure quotients.
This talk is based on joint work with L\'opezPermouth, Tolooei and Ungor.
[1] L\'opezPermouth and Simental, Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra 362. 5669, 2012.
[2] Aydogdu and L\'opezPermouth, An alternative perspective on the injectivity of modules, J. Algebra 338, 207219, 2011.
[3] Harmanci, L\'opezPermouth, and Ungor, On the pureinjectivity profile of a ring. preprint.