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An approach to extending and lifting modules by modular lattices. (English) Zbl 0998.16004
The aim of this paper is twofold: first, to place a module-theoretical result on extending modules into a latticial setting, and second, by applying this lattice-theoretical result to the dual lattice of the lattice of all submodules of a module, to obtain a result on lifting modules. More precisely, the following result is proved. Let $$M$$ be right module over an associative ring with identity element which is the direct sum of two submodules $$M_1$$ and $$M_2$$. Then $$M$$ is a lifting module if and only if $$M$$ is amply supplemented and every coclosed submodule $$N$$ of $$M$$ with $$M=N+M_1$$ and $$M=N+M_2$$ is a direct summand of $$M$$. As a consequence, it follows that if a module $$M$$ is a direct sum of finitely many relatively projective modules $$M_i$$, $$i=1,\dots,n$$, then $$M$$ is a lifting module if and only if $$M$$ is amply supplemented and each $$M_i$$ is lifting.
Reviewer: Toma Albu (Ankara)

##### MSC:
 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 06C05 Modular lattices, Desarguesian lattices 16D80 Other classes of modules and ideals in associative algebras 16D40 Free, projective, and flat modules and ideals in associative algebras