Can the direct product of a family of groups/modules be regarded as the direct limit of a directed system? If yes, than can someone elaborate the desired directed system with it's objects and maps?
2026-03-25 23:58:39.1774483119
Direct product of a family of groups/modules is the direct limit of a directed system formed by the family of groups/modules?
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Inverse limit. Let $\{M_i\}_{i\in I}$ be a family of $R$-modules. The ordered set $(I,=)$ with is right directed and it induces a natural inverse system $\{M_i,1_{M_i}\}_{i\in I}$ which inverse limit $\varprojlim M_i$ is trivially isomorphic to $\prod_{i\in I} M_i$ (both satisfies the same universal property).