Σ-pure injectivity and Brown representability

2013/04/03 by Simion Breaz, Breaz, Simion
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.KT #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.1304.0979

6 pages; comments are welcome! v4: minor revision; the title was changed; accepted by Proc. Amer. Math. Soc

arxiv created 2014/02/20 · arxiv updated 2014/02/21

Abstract

We prove that a right R-module M is Σ-pure injective if and only if Add(M)⊆ Prod(M). Consequently, if R is a unital ring, the homotopy category K(Mod- R) satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply the main result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman.

Related

• Coneat Injective Modules
• Σ-pure-injectives in homotopy categories for gentle algebras
• The Existence of Relative pure Injective Envelopes
• Relative injective modules, superstability and noetherian categories
• A note on pure-semisimple rings