有限生成模与内射模的子模闭

SUBMODULE CLOSURE OF FINITELY GENERATEI) MODULE AND INJECTIVE MODULE

本文提出有限生成模与内射模的子模闭的充要条件,即:每一有限生成R—模的子模仍为有限生成模的充要条件是:R的每一左理想均为有限生成R—模。每一内射R—模的子模仍为内射模的充要条件是:R的每一左理想均为内射R—模。

This paper discusses some necessary and sufficicnt conditions aboutsubmodule closure of finitely generated module and injective module. Each ofthem is as follows. Every submodule of finitely generated R-module is also finitely generatedif and only if every left ideal of R is a finitely generated R-module. Every submodule of injective R-module is also injective if and only ifevery left ideal of R is an injective R-module.