 Tensor product of modules

For a right module $M_R$ and a left module ${}_{R}N$, their tensor product of modules is the abelian group generated by symbols $m\otimes n$, modulo additivity in each variable andThe universal property of the tensor product of modules identifies additive maps from it with balanced maps on $M\times N$.

When $R$ is a commutative ring, two left modules can be used by viewing the first as a right module. The tensor product is then an $R$-module with $r(m\otimes n)=(rm)\otimes n=m\otimes (rn)$. For example, $\mathbb{Z}/a\mathbb{Z}{\otimes }_{\mathbb{Z}}\mathbb{Z}/b\mathbb{Z}\cong \mathbb{Z}/\gcd (a,b)\mathbb{Z}$: the generator $1\otimes 1$ is killed by both $a$ and $b$, and the universal property identifies exactly those relations. For a noncommutative ring, the left and right input actions must be specified.

In algebra, the tensor product is a way to construct a new module from two given modules, effectively allowing us to "multiply" the modules together. It is particularly useful in the context of linear algebra, representation theory, and algebraic topology. ### Definition Let \( R \) be a ring, and let \( M \) and \( N \) be two \( R \)-modules.