- Hacettepe Journal of Mathematics and Statistics
- Vol: 45 Issue: 3
- Small supplements, weak supplements and proper classes
Small supplements, weak supplements and proper classes
Authors : Rafail Alizade, Engin Büyükasik, Yılmaz Durğun
Pages : 649-661
View : 12 | Download : 3
Publication Date : 2016-06-01
Article Type : Research
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 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) $\cap$ 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