an:01724044
Zbl 0998.16004
Keskin, Derya
An approach to extending and lifting modules by modular lattices
EN
Indian J. Pure Appl. Math. 33, No. 1, 81-86 (2002).
00083382
2002
j
16D70 06C05 16D80 16D40
modular lattices; lattices of submodules; extending modules; lifting modules; direct sums; amply supplemented modules; coclosed submodules; direct summands; relatively projective modules
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.
Toma Albu (Ankara)