1. The problem is to define the tensor product of modules. 2. The tensor product of modules is a construction in abstract algebra that combines two modules over a commutative ring into a new module. 3. Given two modules $M$ and $N$ over a commutative ring $R$, the tensor product $M \otimes_R N$ is an $R$-module together with a bilinear map $\otimes : M \times N \to M \otimes_R N$ satisfying the universal property: for any $R$-module $P$ and any bilinear map $f : M \times N \to P$, there exists a unique $R$-module homomorphism $\tilde{f} : M \otimes_R N \to P$ such that $f = \tilde{f} \circ \otimes$. 4. Important rules: - Bilinearity means $\otimes$ is linear in each argument separately. - The tensor product is constructed by taking the free module generated by symbols $m \otimes n$ for $m \in M$, $n \in N$, and then factoring out the submodule generated by relations enforcing bilinearity. 5. In simpler terms, the tensor product $M \otimes_R N$ is a module that encodes all possible products of elements from $M$ and $N$ subject to linearity conditions. 6. This construction is fundamental in many areas of algebra, including homological algebra and representation theory.