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

Definition (free module over a set):

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

Failed to parse (unknown function "\middle"): {\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

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

and the obvious addition

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

Definition (tensor product):

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

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

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

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

Proposition (universal property of the tensor product):

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

${\displaystyle }$.

Proposition (tensor product as multifunctor):

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

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

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

Failed to parse (unknown function "\begin{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 $R$ be a ring