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

No. Take $R=\mathbb{Z}$, $M=\mathbb{Q}$, andEvery $N_i$ is a torsion group, so localization at the nonzero integers givesand hence ${\prod }_i(\mathbb{Q}{\otimes }_{\mathbb{Z}}{N}_i)=0$.

By contrast, the element $x=(1\mathrm{mod}i{)}_{i\ge 2}$ of ${\prod }_i{N}_i$ has infinite order: no positive integer is divisible by every $i$. The canonical descriptionshows that $1\otimes x$ is nonzero, because an element dies in this localization only if one nonzero integer annihilates it. Thus the left module is nonzero while the right module is zero, giving the tensor product and infinite direct product counterexample.

Solved by gpt-5.6-sol high.