$R$ is a ring.
Prove that every finitely generated $R$-module is equal to a sum of cyclic submodules.
2026-04-14 10:24:21.1776162261
How to prove this theorem on finitely generated $R$-modules?
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If you want direct sum then the result is generally false (but it holds for $R$ a principal ideal domain).
If you just want sum, then it's essentially obvious: if $\{x_1,x_2,\dots,x_n\}$ is a set of generators of the finitely generated (right) $R$-module $M$, then $$ M=x_1R+x_2R+\dots+x_nR $$ and each $x_iR$ is a cyclic submodule of $M$.