Chinese remainder for modules

Module quotients by comaximal ideal multiples split as a direct sum of smaller quotients.

Chinese remainder for modules

Chinese remainder for modules: Let $R$ be a commutative ring, let $M$ be an $R$ -module, and let $I_1,\dots,I_n\subset R$ be pairwise comaximal ideals. Then the canonical map

M\Big/\Bigl(\bigcap_{j=1}^n I_j M\Bigr)\longrightarrow \bigoplus_{j=1}^n M/I_j M

is an isomorphism of $R$ -modules. In particular, if $I+J=R$ then $M/(IM\cap JM)\cong M/IM\oplus M/JM$ .

This is the module-level consequence of the Chinese remainder theorem for pairwise comaximal ideals in a commutative ring , expressed as an isomorphism of quotient modules into a direct sum .