- 10.1 Definition of a module
- 10.2 Direct sums and products
- 10.3 Chain conditiions
- 10.4 Composition series
- 10.5 Semisimple modules
- 10.6 Tensor products
- 10.7 Modules over a principal ideal domain

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: R × M -> M denoted by (r,m)=rm, for all rR and all m M,

such that for all r,r

(i) r(m

(ii) ( r

(iii) r

(iv) 1 m = m .

** Example.** 10.1.1.
(Vector spaces over F are F-modules)
If V is a vector space over a field F,
then it is an abelian group under addition of vectors.
The familiar rules for scalar multiplication
are precisely those needed to show that V
is a module over the ring F.

** Example.** 10.1.2.
(Abelian groups are * Z*-modules)
If A is an abelian group with its operation denoted additively,
then for any element xZ
and any positive integer n,
we have defined nx to be the sum of x with itself n times.
This is extended to negative integers by taking sums of -x.
With this familiar multiplication,
it is easy to check that A becomes a

Another way to show that A is a * Z*-module
is to define a ring homomorphism
:

If M is a left R-module,
then there is an obvious definition of a
** submodule**
of M: any subset of M
that is a left R-module under the operations induced from M.
The subset {0} is called the
** trivial submodule**,
and is denoted by (0).
The module M is a submodule of itself, an
** improper**
submodule.
It can be shown that if M is a left R-module, then a subset
NM
is a submodule if and only if it is nonempty, closed under sums,
and closed under multiplication by elements of R.

If N is a submodule of _{R}M,
then we can form the factor group M/N.
There is a natural multiplication defined on the cosets of N: for any
rR and any
xM,
let r(x+N)=rx+N.
If x+N=y+N,
then x-yN,
and so rx-ry=r(x-y)N,
and this shows that scalar multiplication is well-defined.
It follows that M/N is a left R-module.

Any submodule of _{R}R is called a
** left ideal**
of R.
A submodule of R_{R} is called a
** right ideal**
of R, and it is clear that a subset of R is an ideal
if and only if it is both a left ideal and a right ideal of R.

For any element m of the module M, we can construct the submodule

Rm = { x M | x = rm for some r R }.

This is the smallest submodule of M that contains m, so it is called the
< X > =
_{xX}Rx.

M=Rm_{i}.

The module M is called a

m= a

We note that if N is a submodule of M such that
N and M/N are finitely generated,
then M is finitely generated.
In fact, if
x_{1},
x_{2}, . . . ,
x_{n} generate N and
y_{1}+N,
y_{2}+N, . . . ,
y_{m}+N generate M/N, then
x_{1}, . . . ,
x_{n},
y_{1}, . . . ,
y_{m} generate M.

The module _{R}R
is the prototype of a free module, with generating set {1}.
If _{R}M is a module,
and XM,
we say that the set X is
** linearly independent** if
a_{i}
x_{i}=0 implies
a_{i}=0 for i=1,...,n,
for any distinct x_{1},
x_{2}, . . . ,
x_{n}
X and any
a_{1},
a_{2}, . . . ,
a_{n}
R.
Then a linearly independent generating set for M is called a
** basis**
for M, and so M is a free module
if and only if it has a basis.

** 10.1.3 Definition.**
Let M and N be left R-modules.
A function f:M -> N is called an
** R-homomorphism**
if

f(m_{1} +
m_{2}) =
f(m_{1}) +
f(m_{2}) and
f(rm) = rf(m)

Hom_{R}(M,N) or
Hom(_{R}M,_{R}N).

ker(f) = { m M | f(m) = 0 }.

We say that f is an
End_{R}(M).

** 10.1.5 Theorem.**
Let N, N_{0},
M_{0} be submodules of
_{R}M.

** (a)**
N_{0} /
(N_{0}
M_{0})
(N_{0} +
M_{0}) /
M_{0}.

** (b)**
If N_{0}
N, then
(M / N_{0}) /
(N / N_{0})
M / N.

** (c)**
If N_{0}
N, then
N
(N_{0} +
M_{0}) =
N_{0} +
(N
M_{0}).

** 10.1.6 Lemma.**
Let X be any subset of the module _{R}M.
Any submodule N with
NX(0)
is contained in a submodule maximal with respect to this property.

A submodule N of the left R-module M is called a
** maximal submodule**
if NM and for any submodule K with
NKM,
either N=K or K=M.
Consistent with this terminology,
a left ideal A of R is called a
** maximal left ideal**
if AR and for any left ideal B with
ABR,
either A=B or B=R.
Thus A is maximal precisely when it is a maximal element
in the set of proper left ideals of R, ordered by inclusion.
It is an immediate consequence of Lemma 10.1.6
that every left ideal of the ring R is contained in a maximal left ideal,
by applying the proposition to the set X = {1}.
Furthermore, any left ideal maximal with respect to not including 1
is in fact a maximal left ideal.

** 10.1.7 Proposition.**
For any nonzero element m of the module
_{R}M
and any submodule N of M with mN,
there exists a submodule N* maximal with respect to
N*N and
mN*.
Moreover, M/N* has a minimal submodule
contained in every nonzero submodule.

** 10.1.8 Corollary.**
Any proper submodule of a finitely generated module
is contained in a maximal submodule.

** 10.1.9 Definition.**
Let R be a ring, and let M be a left R-module.
For any element mM, the left ideal

Ann(m) = { r R | r m = 0 }

is called theAnn (M) = { r R | r m = 0 for all m M }.

is called theThe module M is called

** 10.1.10 Definition.**
A nonzero module _{R}M is called
** simple**
(or ** irreducible**)
if its only submodules are (0) and M.

We first note that a submodule
NM
is maximal if and only if M/N is a simple module.
A submodule NM is called a
** minimal submodule**
if N(0) and for any submodule K with
NK(0),
either N=K or K=(0).
With this terminology,
a submodule N is minimal if and only if it is simple
when considered as a module in its own right.

** 10.1.11 Lemma. [Schur]**
If _{R}M is simple, then
End_{R}(M) is a division ring.

** 10.1.12 Proposition.**
The following conditions hold for a left R-module M.

**(a)**
The module M is simple if and only if Rm=M, for each nonzero
mM.

**(b)**
If M is simple, then Ann(m) is a maximal left ideal,
for each nonzero mM.

**(c)**
If M is simple, then it has the structure
of a left vector space over a division ring.

The submodule of
_{ I}M_{}
consisting of all elements m such that
m_{}=0
for all but finitely many components
m_{}
is called the
** direct sum**
of the modules
{M_{}}_{ I},
and is denoted by
_{ I}M_{}.

** 10.2.2 Proposition.**
Let {M_{}}_{ I}
be a collection of left R-modules
indexed by the set I, and let N be a left R-module.

For each
I
let
p_{}:_{ I}M_{}->M_{}
be the projection defined by
p_{}(m)=m_{},
for all
m_{ I}M_{},
and let
i_{}:M_{}->_{ I}M_{}
be the inclusion defined for all
xM_{}
by i_{}(x)=m,
where
m_{}=x and
m_{i}=0 for all
i.

**(a)**
For any set
{f_{}}_{ I}
of R-homomorphisms such that
f_{}:N->M_{}
for each
I,
there exists a unique R-homomorphism
f:N->_{ I}M_{}
such that
p_{}f=f_{}
for all
I.

**(b)**
For any set
{f_{}}_{ I}
of R-homomorphisms such that
f_{}:M_{}->N
for each
I,
there exists a unique R-homomorphism
f:_{ I}M_{}->N
such that
fi_{}=f_{}
for each
I.

** 10.2.3 Proposition.**
Let M be a left R-module.

**(a)**
The module M is free if and only if
it is isomorphic to a direct sum
R^{I},
for some index set I.

**(b)**
The module M is a homomorphic image
of a free module.

** 10.2.4 Proposition.**
Let M and M_{1}, . . . ,
M_{n} be left R-modules.
Then

M
M_{1}
M_{2}
^{ . . . }
M_{n}

if and only if there exist R-homomorphisms
i_{j}:M_{j}->M
and
p_{j}:M->M_{j}
for j=1, . . . , n such that

p_{j}
i_{k} =
_{j}_{k}
and
i_{1}p_{1}
+ . . . +
i_{n}p_{n}
= 1_{M}.

** 10.2.5 Proposition.**
Let A_{1},
A_{2}, . . . ,
A_{n} be left ideals of the ring R.

** (a)**
R = A_{1}
A_{2}
^{ . . . }
A_{n}
if and only if there exists a set
e_{1},
e_{2}, . . . ,
e_{n}
of orthogonal idempotent elements of R such that
A_{j}=Re_{j}
for
1jn
and
e_{1} +
e_{2} + . . . +
e_{n} = 1.

** (b)**
The left ideals A_{j} in part (a)
are two-sided ideals if and only if
the corresponding idempotent elements belong to the center of R.

** (c)**
If condition (b) holds,
then every left R-module M can be written as a direct sum

M=M_{1}
M_{2}
^{ . . . }
M_{n},
where
M_{j} is a module over the ring
A_{j},
for
1jn.

** 10.2.6 Definition.**
Let L, M, N be left R-modules.

An onto R-homomorphism f:M->N is said to be
** split**
if there exists an R-homomorphism g:N->M with
fg=1_{N}.

A one-to-one R-homomorphism g:L->M is said to be
** split**
if there exists an R-homomorphism f:M->L such that
fg=1_{L}.

** 10.2.7 Proposition.**
Let M, N be left R-modules.

** (a)**
Let f:M->N and g:N->M
be R-homomorphisms such that
fg=1_{N}.
Then M=ker(f)Im(g).

** (b)**
A one-to-one R-homomorphism g:N->M splits
if and only if Im(g) is a direct summand of M.

** (c)**
An onto R-homomorphism f:M->N splits
if and only if ker(f) is a direct summand of M.

** 10.2.8 Proposition.**
Let L, M, and N be left R-modules.
Let g:L->M be a one-to-one R-homomorphism,
and let f:M->N be an onto R-homomorphism
such that Im(g)=ker(f).
Then g is split if and only if f is split,
and in this case
MLN.

** 10.2.9 Corollary.**
The following conditions are equivalent for the module
_{R}M:

** (1)**
every submodule of M is a direct summand;

** (2)**
every one-to-one R-homomorphism into M splits;

** (3)**
every onto R-homomorphism out of M splits.

** 10.2.10 Definition.**
A module _{R}M is called
** completely reducible**
if every submodule of M is a direct summand of M.

** 10.2.11 Proposition.**
The following conditions are equivalent
for the module _{R}P:

** (1)**
every R-homomorphism onto P splits;

** (2)**
P is isomorphic to a direct summand of a free module;

** (3)**
for any onto R-homomorphism p:M->N
and any R-homomorphism f:P->N there exists a lifting
f*:P->M such that pf*=f.

** 10.2.12 Definition.**
A module _{R}M is called
** projective**
if it is isomorphic to a direct summand of a free module.

M

of submodules of M must terminate after a finite number of steps.

Similarly, M is said to be

M

of submodules of M must terminate after a finite number of steps.

** 10.3.2 Definition.**
A ring R is said to be
** left Noetherian**
if the module _{R}R is Noetherian.

A ring R is said to be
** left Artinian**
if the module _{R}R is Artinian.

If R satisfies the conditions for both right and left ideals,
then it is simply said to be
** Noetherian**
or ** Artinian**.

** 10.3.3 Proposition.**
The following conditions are equivalent for a module
_{R}M:

** (1)**
M is Noetherian;

** (2)**
every submodule of M is finitely generated;

** (3)**
every nonempty set of submodules of M has a maximal member.

** 10.3.4 Proposition.**
The following conditions hold for a module
_{R}M and any submodule N.

** (a)**
M is Noetherian if and only if N and M/N
are Noetherian.

** (b)**
M is Artinian if and only if N and M/N
are Artinian.

** 10.3.5 Corollary.**
A finite direct sum of modules is Noetherian
if and only if each summand is Noetherian;
it is Artinian if and only if each summand is Artinian.

** 10.3.6 Proposition.**
A ring R is left Noetherian if and only if
every finitely generated left R-module is Noetherian;
it is left Artinian if and only if
every finitely generated left R-module is Artinian.

** 10.3.7 Theorem.
[Hilbert basis theorem]
**
If R is a left Noetherian ring, then so is the polynomial ring R[x].

** 10.3.8 Definition.**
Let D be a principal ideal domain.
and left M be a D-module.
We say that M is a
** torsion module**
if Ann(m)(0)
for all nonzero elements mM.

** 10.3.9 Proposition.**
Let D be a principal ideal domain.
Any finitely generated torsion D-module has finite length.

We can now give some fairly wide classes of examples
of Noetherian and Artinian rings.
If D is a principal ideal domain,
then D is Noetherian since each ideal is generated by a single element.
It follows that the polynomial ring
D[x_{1},x_{2},...,x_{n}]
is also Noetherian.
If F is a field, then F[x]/I is Artinian,
for any nonzero ideal I of F[x],
since F[x] is a principal ideal domain.
This allows the construction of many interesting examples.
Note that D[x]/I need not be Artinian
when D is assumed to be a principal ideal domain rather than a field, since
* Z*[x]/<x> is isomorphic to