# Linear Algebra over a Ring/Direct product, direct sum and tensor product

Definition (free module over a set):

Let ${\displaystyle S}$ be any set, and let ${\displaystyle R}$ be a ring. Then the free module ${\displaystyle F(S)}$ is defined to be the module

$\displaystyle F(S) := \left\{ \sum_{k=1}^n r_k s_k \middle| \forall k \in [n]: r_k \in R \wedge s_k \in S \right\}$

together with the module operation

${\displaystyle r\left(\sum _{k=1}^{n}r_{k}s_{k}\right)=\sum _{k=1}^{n}rr_{k}s_{k}}$

${\displaystyle \sum _{k=1}^{n}r_{k}s_{k}+\sum _{k=1}^{n}r_{k}'s_{k}'=\sum _{k=1}^{n}r_{k}s_{k}+\sum _{k=1}^{n}r_{k}'s_{k}'}$.

Definition (tensor product):

Let ${\displaystyle R}$ be a ring and let ${\displaystyle M_{1},\ldots ,M_{n}}$ be ${\displaystyle R}$-modules. The tensor product of the modules ${\displaystyle M_{1},\ldots ,M_{n}}$ is defined as the ${\displaystyle R}$-module

${\displaystyle M_{1}\otimes M_{2}\otimes \cdots \otimes M_{n}:=F(M_{1}\times M_{2}\times \cdots \times M_{n})/K}$,

where ${\displaystyle K\leq F(M_{1}\times M_{2}\times \cdots \times M_{n})}$ is the following submodule:

{\displaystyle {\begin{aligned}K=&\langle {\big \{}(m_{1},\ldots ,m_{j-1},m_{j}-rl,m_{j+1},\ldots ,m_{n})-(m_{1},\ldots ,m_{n})-r(m_{1},\ldots ,m_{j-1},l,m_{j+1},\ldots ,m_{n})\\&{\big |}j\in \{1,\ldots ,n\},m_{1}\in M_{1},\ldots ,m_{n}\in M_{n},l\in M_{j},r\in R{\big \}}\rangle \end{aligned}}}.

Proposition (universal property of the tensor product):

Let ${\displaystyle R}$ be a ring and let ${\displaystyle M_{1},\ldots ,M_{n}}$ be ${\displaystyle R}$-modules. Then the tensor product ${\displaystyle M_{1}\otimes \cdots \otimes M_{n}}$ satisfies the universal property that for each ${\displaystyle R}$-module ${\displaystyle N}$ and each multilinear map ${\displaystyle f:M_{1}\times \cdots \times M_{n}\to N}$, there exists a unique linear map ${\displaystyle {\tilde {f}}:M_{1}\otimes \cdots \otimes M_{n}\to N}$ such that

.

{{proposition|tensor product as multifunctor|Let ${\displaystyle R}$ be a ring. Then for each ${\displaystyle n\in \mathbb {N} }$, the tensor product yields a multifunctor

${\displaystyle \oplus _{n}:(R-{\textbf {Mod}})^{n}\to R-{\textbf {Mod}}}$.

Whenever ${\displaystyle M_{1},\ldots ,M_{n}}$ and ${\displaystyle N_{1},\ldots ,N_{n}}$ are ${\displaystyle R}$-modules and for ${\displaystyle j\in \{1,\ldots ,n\}}$, ${\displaystyle f_{j}:M_{j}\to N_{j}}$ are morphisms, the morphisms that turn ${\displaystyle \oplus _{n}}$ into a multifunctor are given by

align}"): {\displaystyle \begin{align} f_1 \otimes \cdots \otimes f_n: M_1 \otimes \cdots \otimes M_n \to N_1 \otimes \cdots \otimes}} {{proposition|associativity of the tensor product|Let [itex]R be a ring [itex]