Definition (free module over a set):
Let
be any set, and let
be a ring. Then the free module
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
![{\displaystyle r\left(\sum _{k=1}^{n}r_{k}s_{k}\right)=\sum _{k=1}^{n}rr_{k}s_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7081d8d48a172a738153b8a2f6077156c71bf1f0)
and the obvious addition
.
Definition (tensor product):
Let
be a ring and let
be
-modules. The tensor product of the modules
is defined as the
-module
,
where
is the following submodule:
.
Proposition (tensor product as multifunctor):
Let
be a ring. Then for each
, the tensor product yields a multifunctor
.
Whenever
and
are
-modules and for
,
are morphisms, the morphisms that turn
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
be a ring