For $R$-modules $M,N$, the <tensor product of modules> is an $R$-module $M\otimes_RN$ together with the balanced map
$$
\tau:M\times N\longrightarrow M\otimes_RN,
\qquad
(m,n)\longmapsto m\otimes n,
$$
such that every <balanced map> $b:M\times N\to P$ factors through one unique $R$-linear map $\widetilde b$:
$$
b=\widetilde b\circ\tau.
$$
Equivalently,
$$
\operatorname{Hom}_R(M\otimes_RN,P)
\cong\{\text{balanced maps }M\times N\to P\}
$$
naturally in $P$. This is the <universal property of the tensor product of modules>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-101/1/i/solution