Exercises on tensor product

$\Z/(m)\otimes_{\Z} \Z/(n)=\Z/((m,n))$.

1. $\otimes\colon ([a]_m,[b]_n)\to[ab]_{(m,n)}$ is a well defined linear map from $\Z/(m)\times \Z/(n)$ to $\Z/((m,n))$.

2.For any bilinear operator $\phi\colon \Z/(m)\times \Z/(n)\to M$, $\tilde{\phi}([1]_{(m,n)})=\phi([1]_m,[1]_n)$ defines a linear operator from $\Z/((m,n))$ to $M$. This is because $(m,n)\phi([1]_m,[1]_n)=(mx+ny)\phi([1]_m,[1]_n)=\phi([mx]_m,[1]_n)+\phi([1]_m,[ny]_n)=\phi([0]_m,[1]_n)+\phi([1]_m,[0]_n)=0,$ where $mx+ny=(m,n)$.

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