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 (\sum_{k = 1}^{n} r_{k} s_{k}) = \sum_{k = 1}^{n} r r_{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 $M_{1}, \dots, M_{n}$ be $R$ -modules. The tensor product of the modules $M_{1}, \dots, M_{n}$ is defined as the $R$ -module

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

,

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

\begin{matrix} K = & ⟨ {(m_{1}, \dots, m_{j - 1}, m_{j} - r l, m_{j + 1}, \dots, m_{n}) - (m_{1}, \dots, m_{n}) - r (m_{1}, \dots, m_{j - 1}, l, m_{j + 1}, \dots, m_{n}) \\ | j \in {1, \dots, n}, m_{1} \in M_{1}, \dots, m_{n} \in M_{n}, l \in M_{j}, r \in R} ⟩ \end{matrix}

.

Proposition (universal property of the tensor product):

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

.

Proposition (tensor product as multifunctor):

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

\oplus_{n} : (R - 𝐌𝐨𝐝)^{n} \to R - 𝐌𝐨𝐝

.

Whenever $M_{1}, \dots, M_{n}$ and $N_{1}, \dots, N_{n}$ are $R$ -modules and for $j \in {1, \dots, n}$ , $f_{j} : M_{j} \to N_{j}$ 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