Definition (multilinear function):
Let
be a ring, and let
be
-modules. Then the set of
-multilinear functions from
to
is the set
.
Proposition (equivalent definition of tensor product of free modules using multilinear functions):
Let
be a ring, and let
be free, finitely generated
-modules. Then if we alternatively define
,
and let the elementary tensors be
, then the
from this definition satisfies the same universal property as the usual tensor product
. In particular, the two tensor products are canonically isomorphic.
Proof: For
, let
be a basis of
, where
is the respective finite index set. Given any
-module
and any multilinear map
, we want a unique linear function
such that
, where
is the map that sends a tuple to the respective elementary tensor.