An Isomorphism

Groups, Rings, Domains, Modules, etc, Galois theory
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Tsakanikas Nickos
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An Isomorphism

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Post by Tsakanikas Nickos »

Let \( A \) be a ring (commutative and associative ring with unity), let \( \mathfrak{a} \) be an ideal of \( A \) and let \( M \) be an \( A \)-module. Show that \[ \displaystyle \left( A/ \mathfrak{a} \right) \otimes_{A} M \cong M / \mathfrak{a} M \]where \( \mathfrak{a} M \) is the submodule of \( M \) defined as the set of all finite sums \( \Sigma a_{i}m_{i} \), where \( a_{i} \in A \, , \, m_{i} \in M \).
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