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OUOSU ring theory seminar

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Pure-injectivity from an alternative perspective, by Joseph Mastromatteo (Ohio University)

What
  • Seminar
When Apr 04, 2014
from 04:45 PM to 05:45 PM
Where CH 240 (OSU, Columbus)
Contact Name
Contact Phone 740 593 1258
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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 N-injective). 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 so-called subinjectivity domain of M. M is N-subinjective 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 pure-injectivity 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 pure-subinjectivity. A module M is said to be N-pure-subinjective 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 pure-subinjectivity domains with the other domains of pure-injectivity and injectivity. We also consider when the pure-subinjectivity domain of a module is closed under pure quotients.

This talk is based on joint work with L\'opez-Permouth, Tolooei and Ungor.

 

[1] L\'opez-Permouth and Simental, Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra 362. 56-69, 2012.

[2] Aydogdu and L\'opez-Permouth, An alternative perspective on the injectivity of modules, J. Algebra 338, 207-219, 2011.

[3] Harmanci, L\'opez-Permouth, and Ungor, On the pure-injectivity profile of a ring. preprint.

 

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