Small supplements, weak supplements and proper classes

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Alizade R. , Buyukasik E., Durgun Y.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.45, no.3, pp.649-661, 2016 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 45 Issue: 3
  • Publication Date: 2016
  • Doi Number: 10.15672/hjms.20164512507
  • Page Numbers: pp.649-661


Let SS denote the class of short exact sequences E :0 -> A (f) under right arrow B -> C -> 0 of R-modules and R-module homomorphisms such that f(A) has a small supplement in B i.e. there exists a submodule K of M such that f(A) + K = B and f(A) boolean AND K is a small module. It is shown that, SS is a proper class over left hereditary rings. Moreover, in this case, the proper class SS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplement submodules. The homological objects, such as, SS-projective and SS-coinjective modules are investigated. In order to describe the class SS, we investigate small supplemented modules, i.e. the modules each of whose submodule has a small supplement. Besides proving some closure properties of small supplemented modules, we also give a complete characterization of these modules over Dedekind domains.