Small supplements, weak supplements and proper classes

Authors : Rafail ALİZADE, Engin BÜYÜKASIK, Yılmaz DURĞUN

Pages : 649-661

View : 52 | Download : 9

Publication Date : 2016-06-01

Article Type : Research Paper

Abstract :Let  $ \mathcal{S} \mathcal{S}  $ denote the class of short exact sequences $E :0 \to A \ bar{f} \to  C\to 0 $ of R-modules and R-module homomorphisms such that finsert ignore into journalissuearticles values(A); has a small supplement in B i.e. there exists a submodule K of M such that finsert ignore into journalissuearticles values(A); +K = B and finsert ignore into journalissuearticles values(A); $\cap$ 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, $\mathcal SS$-projective and $\mathcal SS$- coinjective modules are investigated. In order to describe the class $\mathcal SS$, 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 ove r Dedekind domains.
Keywords : Proper class of short exact sequences, weak supplement submodule, small module, small supplement submodule

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