I have a ring $R$, an ideal $I$ of $R$ and an $R$-module $M$. How does one define $IM$. Is it the set $\{im\ \vert\ i\in I, m\in M\}$ or is it the smallest module containing these elements?
2026-05-16 13:04:07.1778936647
product of a module by an ideal
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Assuming that $R$ is commutative, then we can proceed as in A&M page 19.
Let $I\subset R$ be an ideal, and let $M$ be an $R$-module. Then, $IM$ is the set of all finite sums $\sum r_ix_i$ where $r_i\in I$ and $x_i\in M$. This is of course a submodule of $M$.