Small supplements, weak supplements and proper classes

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 over Dedekind domains.

Keywords

• Proper class of short exact sequences,weak supplement submodule,small module,small supplement submodule

References

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