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Commutative algebra
Yoshifumi Tsuchimoto
DEFINITION 01.1
A (unital associative)
ring is a set
equipped with two binary operations
(addition (``+'') and multiplication (``
'')) such that
the following axioms are satisfied.
 Ring1.

is an additive group with respect to the addition.
 Ring2.
 distributive law holds. Namely, we have
 Ring3.
 The multiplcation is associative.
 Ring4.

has a multiplicative unit.
In this lectuer we are mainly interested in commutative rings,
that means, rings on which the multiplication satisfies the commutativity law.
For any ring
, we denote by
(respectively,
)
the zero element of
(respectively, the unit element of
).
Namely,
and
are elements of
characterized by
the following rules.
When no confusion arises, we omit the subscript `
'
and write
instead of
.
DEFINITION 01.2
A map
from a unital associative ring
to another unital associative ring
is said to be
ring homomorphism if it satisfies the following conditions.
 Ringhom1.

 Ringhom2.

 Ringhom3.

DEFINITION 01.5
Let
be modules over a ring
.
Then a map
is called an
module homomorphism if
it is additive and preserves the
action.
The set of all module homomorphisms from
to
is denoted by
. It has an structure of an module in an obvious manner.
Furthermore, when
is a commutative ring, then it has a structure of
an
module.
DEFINITION 01.6
An subset
of an
module
is said to be an
submodule of
if
itself is an
module and the inclusion map
is an
module homomorphism.
DEFINITION 01.7
An subset
of an
module
is said to be an
submodule of
if
itself is an
module and the inclusion map
is an
module homomorphism.
DEFINITION 01.8
Let
be a ring. Let
be an
submodule of an
module
.
Then we may define the
quotient
by
where the equivalence relation
is defined as follows:
It may be shown that the quotient
so defined is actually an
module
and that the natural projection
is an
module homomorphism.
DEFINITION 01.9
Let
be an
module homomorphism between
modules.
Then we define its
kernel as follows.
The kernel and the image of an
module homomorphism
are
modules.
THEOREM 01.10
Let
be an
module homomorphism between
modules.
Then
DEFINITION 01.11
Let
be a ring.
An ``sequence''
is said to be
an exact sequence of
modules
if the following conditions are satisfied
 Exact1.

are
modules.
 Exact2.

are
module homomorphisms.
 Exact3.

.
For any
submodule
of an
module
, we have the
following exact sequence.
EXERCISE 01.1
Compute the following modules.

.

.

.
DEFINITION 01.12
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

.

.

.
DEFINITION 01.14
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
EXERCISE 01.2
Compute
and
.
DEFINITION 01.16
A left
module
is said to be
flat if
is an exact functor.
Namely, for any exact sequence
of left
modules, the sequence
is also exact.
**The following two facts may give some intuitive idea of what flatness means.
THEOREM 01.17
If
is a Noetherian ring and
is a finitelygenerated
module,
then
is flat over
if and only if the associated sheaf
on
is locally free.
THEOREM 01.18
[1, Theorem 23.1+Theorem 15.1]
Let
be a regular local ring.
Let
be a CohenMacaulay local ring.
Let
be a local ring homomorphism.We set
for the fiber ring of
over
.
Then an equality
holds if and only if
is flat over
.
**
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20110414