FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 3, PAGES 141-150

Distributive extensions of modules

A. A. Tuganbaev

Abstract

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Let $X$ be a submodule of a module $M$. The extension $X$Í M is said to be distributive if $X$Ç (Y + Z) = X Ç Y + X Ç Z for any two submodules $Y$ and $Z$ of $M$. We study distributive extensions of modules over not necessarily commutative rings. In particular, it is proved that the following three conditions are equivalent: (1) $X$_A Í M_A is a distributive extension; (2) for any submodule $Y$ of the module $M$, no simple subfactor of the module $X/(X$Ç Y) is isomorphic to any simple subfactor of $Y/(X$Ç Y) (3) for any two elements $x$Î X and $m$Î M, there does not exist a simple factor module of the cyclic module $xA/(X$Ç mA) that is isomorphic to a simple factor module of the cyclic module $mA/(X$Ç mA).

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Last modified: July 22, 2006