The <tensor product> $M\otimes_RN$ is the free abelian group on symbols $(m,n)$ modulo additivity in each variable and the balancing relation $(rm,n)=(m,rn)$. It has the universal property that balanced bilinear maps $M\times N\to P$ correspond uniquely to homomorphisms $M\otimes_RN\to P$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 3 / a / Solution

 Solution (source code)