A simple exercise on tensor products
Posted: Wed Jun 15, 2016 11:10 am
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 \(\displaystyle{d=\rm{gcd}(m,n)}\).
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 \(\displaystyle{d=\rm{gcd}(m,n)}\).