- 10.4 Composition series
- 10.5 Semisimple modules
- 10.6 Tensor products
- 10.7 Modules over a principal ideal domain

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M = M_{0}
M_{1}
. . .
M_{n} = (0)

** 10.4.2. Theorem.
[Jordan-Holder]
**
If a module M has a composition series,
then any other composition series for M is equivalent to it.

As an immediate consequence of the Jordan-Holder theorem,
if a module _{R}M has a composition series,
then all composition series for M must have the same length,
which we denote by
(M).
This is called the
** length**
of the module,
and we simply say that the module has ** finite length**.
Since any ascending chain of submodules
can be refined to a composition series,
(M)
gives a uniform bound on the number of terms
in any properly ascending chain of submodules.
We also note that if
M_{1} and
M_{2} have finite length, then

(M_{1}
M_{2}) =
(M_{1}) +
(M_{2}).

** 10.4.3. Proposition.**
A module has finite length if and only if it is both Artinian and Noetherian.

A module _{R}M is said to be
** indecomposable**
if its only direct summands are (0) and M.
As our first example,
we note that * Z*
is indecomposable as a module over itself,
since the intersection of any two nonzero ideals is again nonzero.
To give additional examples of indecomposable

** 10.4.4. Proposition.**
If _{R}M has finite length,
then there exist finitely many indecomposable submodules
M_{1}, M_{2}, . . . , M_{n} such that

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

M = Im (f^{n}) ker(f^{n}) .

** 10.4.7. Proposition.**
Let M be an indecomposable module of finite length,
and let f_{1}, f_{2} be endomorphisms of M.
If f_{1} + f_{2} is an automorphism,
then either f_{1} or f_{2} is an automorphism.

** 10.4.8. Lemma.**
Let X_{1}, X_{2}, Y_{1}, Y_{2} be left R-modules,
and let f: X_{1} X_{2} -> Y_{1} Y_{2} be an isomorphism.
Let
i_{1} : X_{1} -> X_{1} X_{2} and
i_{2} : X_{2} -> X_{1} X_{2}
be the natural inclusion maps, and let
p_{1} : Y_{1} Y_{2} -> Y_{1} and
p_{2} : Y_{1} Y_{2} -> Y_{2}
be the natural projections.
If p_{1} f i_{1} : X_{1} -> Y_{1} is an isomorphism,
then p_{2} f i_{2} : X_{2} -> Y_{2} is an isomorphism.

** 10.4.9. Theorem.
[Krull-Schmidt]
**
Let
{X_{j}}
and
{Y_{i}}
be indecomposable left R-modules of finite length. If

X_{1}
^{ . . . }
X_{m}
Y_{1}
^{ . . . }
Y_{n},

A semisimple module _{R} M behaves like a vector space
in that any submodule splits off, or equivalently,
that any submodule N has a
** complement**
N' such that N+N'=M and NN'=0.

** 10.5.2. Theorem.**
Any submodule of a semisimple module
has a complement that is a direct sum of minimal submodules.

** 10.5.3. Corollary.**
The following conditions are equivalent for a module _{R} M.

** (1)**
M is semisimple;

** (2)**
Soc (M) = M.

** (3)**
M is completely reducible;

** (4)**
M is isomorphic to a direct sum of simple modules.

** 10.5.4. Corollary.**
Every vector space over a division ring has a basis.

** 10.5.5. Definition.**
The module _{R}Q is said to be
** injective**
if for each one-to-one R-homomorphism
i:_{R}M->_{R}N
and each R-homomorphism f:M->Q
there exists an R-homomorphism f*:N->Q
such that f*i=f.

** 10.5.6. Theorem.**
The following conditions are equivalent for the ring R.

** (1)**
R is a direct sum of finitely many minimal left ideals;

** (2)**
_{R} R is a semisimple module;

** (3)**
every left R-module is semisimple;

** (4)**
every left R-module is projective;

** (5)**
every left R-module is injective;

** (6)**
every left R-module is completely reducible.

** 10.5.7. Corollary.**
Let D be a division ring,
and let R be the ring M_{n}(D)
of all n×n matrices over D.
Then every left R-module is completely reducible.

Let R be a ring, and let G be a group.
The ** group ring**
RG is defined to be a free left R-module
with the elements of G as a basis.
The multiplication on RG is defined by

(
_{wG}
a_{w} w )
(
_{xG}
b_{x} x ) =
_{zG}
c_{z} z
where c_{z} =
_{z=wx}
a_{w}
b_{x}.

The crucial property of a group ring is that it converts
group homomorphisms from G into the group of units of a ring
into ring homomorphisms.
To be more precise, let S be a ring,

let
:G->S^{x}
be a group homomorphism,

and let
:R->Z(S)
be any ring homomorphism.

(Recall that S^{x}
denotes the group of invertible elements of S
and Z(S) denotes the center of S.)

Then there is a unique ring homomorphism
:RG->S such that

(g)=(g)
for all gG and
(r)=(r)
for all rR.

** 10.5.8. Theorem.
[Maschke]
**
Let G be a finite group and let K be a field
such that |G| is not divisible by chr(K).
Then every KG-module is completely reducible.

** 10.5.9. Theorem.
[Baer's criterion]
**
For any left R-module Q,
the following conditions are equivalent.

**(1)**
The module Q is injective;

**(2)**
for each left ideal A of R
and each R-homomorphism f:A->Q
there exists an extension f*:R->Q
such that f*(a)=f(a) for all
aA;

**(3)**
for each left ideal A of R
and each R-homomorphism f:A->Q there exists
qQ such that f(a)=aq, for all
aA.

** 10.5.10. Proposition.**
Let D be a principal ideal domain,
with quotient field Q.

**(a)**
The module _{D}Q is injective.

**(b)**
Let I be any nonzero ideal of D,
and let R be the ring D/I.
Then R is an injective module,
when regarded as an R-module.

(i) ( x

(ii) ( x , y

(iii) ( xr, y ) = ( x , ry )

for all x, x

** 10.6.1. Definition.**
A ** tensor product**
of the modules M_{R} and
_{R}N
is an abelian group T(M,N) and an R-bilinear map
: M × N -> T(M,N)
such that for any abelian group A and any R-bilinear map
: M × N -> A
there exists a unique * Z*-homomorphism
f:T(M,N)->A such that
f=.

The group T(M,N) is usually denoted by M

** 10.6.2. Proposition.**
Let M_{R} and
_{R}N be modules. The tensor product
M_{R}N
is unique up to isomorphism, if it exists.

** 10.6.3. Proposition.**
For any modules M_{R} and
_{R}N, the tensor product
M_{R}N
exists.

** 10.6.4. Proposition.**
Let M, M' be right R-modules,
let N, N' be left R-modules, and let
fHom(M_{R},M'_{R})
and
gHom(_{R}N,_{R}N').

**(a)**
Then there is a unique * Z*-homomorphism
fg:M

with (fg) (xy) = f(x) g(y) for all xM, y N.

** 10.6.5. Proposition.**
Let M_{R} be a right R-module, and let
{N_{}}
_{ I}
be a collection of left R-modules.
Then

M _{R}
(
_{ I}
N_{})
_{ I}
( M _{R}
N_{}).

If U is an S-R-bimodule we use the notation

** 10.6.7. Proposition.**
Let R, S, and T be rings, and let
_{S}U_{R},
_{R}M_{T}
, and
_{S}N_{T}
be bimodules.

**(a)**
The tensor product
U_{R}
M is a bimodule, over S on the left and T on the right.

**(b)**
The set Hom_{S}(U,N) is a bimodule,
over R on the left and T on the right.

** 10.6.8. Proposition.**
Let M be a left R-module, and let N be a left S-module.

**(a)**
R_{R}M
M, as left R-modules.

**(b)**
Hom_{S}(S,N)
N, as left S-modules.

** 10.6.9. Proposition.**
Let
_{S}U_{R}
be a bimodule. For any modules
_{R}M and
_{S}N,
there is an isomorphism

:
Hom_{S}(
U _{R}
M , N) ->
Hom_{R}( M,
Hom_{S} (U,N) ).

** 10.6.10. Corollary.**
Let R and S be rings, and let
:R->S be a ring homomorphism.
Let M be any left R-module. Then
S_{R}M
is a left S-module, and for any left S-module N we have

Hom_{S}(
S _{R}
M, N)
Hom_{R}(M, N).

** 10.7.1. Definition.**
Let D be an integral domain,
and let M be a D-module.

An element m M is called a
** torsion element**
if Ann(m)(0).
The set of all torsion elements of M is denoted by tor(M).

If tor(M)=(0), then M is said to be a
** torsionfree**
module.

If tor(M) = M, then M is said to be a
** torsion**
module.

** 10.7.2. Proposition.**
Let D be an integral domain,
and let M be a D-module.
Then tor(M) is a submodule of M,
and M/tor(M) is a torsionfree module.

** 10.7.3. Lemma.**
Let D be a principal ideal domain
with quotient field Q.
Then any nonzero finitely generated submodule of Q
is free of rank 1.

** 10.7.4. Lemma.**
Let D be a principal ideal domain,
and let M be a finitely generated torsionfree D-module.
If M contains a submodule N such that
N is free of rank 1 and M/N is a torsion module,
then M is free of rank 1.

** 10.7.5. Theorem.**
If D is a principal ideal domain,
then any nonzero finitely generated torsionfree D-module is free.

** 10.7.6. Proposition.**
Let D be a principal ideal domain,
and let M be a finitely generated D-module.
Then either M is torsion,
or tor(M) has a complement that is torsionfree.
In the second case, M=tor(M)N
for a submodule NM
such that N is free of finite rank.

The previous proposition shows that to complete the description of all finitely generated modules over a principal ideal domain we only need to characterize the finitely generated torsion modules. The first step is to show that any finitely generated torsion module can be written as a direct sum of finitely many indecomposable modules, and this is a consequence of the next propositions.

** 10.7.7. Proposition.**
Let D be a principal ideal domain,
and let a be a nonzero element of D. If

a =
p_{1}^{m1}
p_{2}^{m2}
^{. . .}
p_{k}^{mk}
is the decomposition of a into a product of irreducible elements,
then we have the following ring isomorphism.

D/aD
(D/p_{1}^{m1}D)
(D/p_{2}^{m2}D)
^{ . . . }
(D/p_{k}^{mk}D)

** 10.7.9. Theorem.**
Let D be a principal ideal domain,
and let M be a finitely generated D-module.
Then M is isomorphic to a finite direct sum of cyclic submodules
each isomorphic to either D or D/p^{k}D,
for some irreducible element p of D.
Moreover, the decomposition is unique up to the order of the factors.