Linear Algebra over a Ring/Multilinear algebra

Definition (multilinear function):

Let $R$ be a ring, and let $M_{1}, \dots, M_{n}, K$ be $R$ -modules. Then the set of $R$ -multilinear functions from $M_{1} \times \dots \times M_{n}$ to $K$ is the set

L (M_{1}, \dots, M_{n}, K) := {f : M_{1} \times M_{n} \to K | \forall j \in [n] : \forall m_{1} \in M_{1}, \dots, m_{n} \in M_{n}, l_{j} \in M_{j}, r \in R : f (m_{1}, \dots, m_{j} + r l_{j}, \dots,}

.

Proposition (equivalent definition of tensor product of free modules using multilinear functions):

Let $R$ be a ring, and let $M_{1}, \dots, M_{n}$ be free, finitely generated $R$ -modules. Then if we alternatively define

M_{1}^{*} \otimes \dots \otimes M_{n}^{*} := L (M_{1}, \dots, M_{n}, R)

,

and let the elementary tensors be $m_{1}^{*} \otimes \dots \otimes m_{n}^{*} (l_{1}, \dots, l_{n}) := m_{1}^{*} (l_{1}) \dots m_{n}^{*} (l_{n})$ , then the $M_{1} \otimes M_{n}$ from this definition satisfies the same universal property as the usual tensor product $M_{1} \otimes \dots \otimes M_{n}$ . In particular, the two tensor products are canonically isomorphic.

Proof: For $j \in [n]$ , let $(e_{λ}^{j})_{λ \in Λ_{j}}$ be a basis of $M_{j}$ , where $Λ_{j}$ is the respective finite index set. Given any $R$ -module $K$ and any multilinear map $f : M_{1} \times \dots \times M_{n} \to K$ , we want a unique linear function $g : M_{1} \otimes \dots \otimes M_{n} \to K$ such that $g \circ h = f$ , where $h : M_{1} \times \dots \times M_{n} \to M_{1}^{*} \otimes \dots \otimes M_{n}^{*}$ is the map that sends a tuple to the respective elementary tensor. $◻$