DEFINITION 08.1
Let
be an associative unital (but not necessarily commutative) ring.
Let
be a right
-module. Let
be a left
-module.
For any (
-)module
, an map

is called an
-balanced biadditive map if

.

.

.

PROPOSITION 08.2Let
be an associative unital (but not necessarily commutative) ring.
Then for any right
-module
and for any left
-module
, there exists
a (
-)module
together with a
-balanced map

which is universal amoung
-balanced maps.

DEFINITION 08.3
We employ the assumption of the proposition above.
By a standard argument on universal objects, we see that such object is
unique up to a unique isomorphism. We call it
the tensor product of
and
and denote it by

LEMMA 08.4Let
be an associative unital ring. Then:

.

For any
,
is a right exact functor.

For any right ideal
of
and for any
-module
,
we have

In particular, if the ring
is commutative, then for any ideals
of
,
we have

DEFINITION 08.5
For any left
-module
,
the left derived functor
of
is called the Tor functor
and denoted by
.

By definition,
may be computed by using projective resolutions of
.