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Cohomologies.
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.
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 01.4
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 01.5
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.
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
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 01.8
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 01.9
Let
be an
module homomorphism between
modules.
Then
DEFINITION 01.10
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.

.

.

.
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20100415