A simple exercise on tensor products

Groups, Rings, Domains, Modules, etc, Galois theory
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Papapetros Vaggelis
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Joined: Mon Nov 09, 2015 1:52 pm

A simple exercise on tensor products

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Post by Papapetros Vaggelis »

1. Let \(\displaystyle{\left(R,+,\cdot\right)}\) be a commutative ring with unity and \(\displaystyle{I\,,J}\)

be ideals of \(\displaystyle{R}\). Then, there exists an \(\displaystyle{R}\) - module isomorphism

\(\displaystyle{(R/I)\bigotimes_{R} (R/J)\cong R/(I+J)}\).

2. Prove that \(\displaystyle{\mathbb{Z}_{m}\bigotimes_{\mathbb{Z}}\mathbb{Z}_{n}\cong \mathbb{Z}_{d}}\)

as \(\displaystyle{\mathbb{Z}}\) - modules, where \(\displaystyle{d=\rm{gcd}(m,n)}\).
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