Z
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 Z-module.
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: R × M -> M
denoted by (r,m)=rm,
for all
rR and all
m M,
R and all m, m1, m2 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 Z-module.
Another way to show that A is a Z-module
is to define a ring homomorphism
:Z->End(A)
by letting (n)=n1,
for all nZ.
This is the familiar mapping that is used to determine
the characteristic of the ring End(A).
The action of Z on A determined by this mapping
is the same one used in the previous paragraph.
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
N
M
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 RM,
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 RR is called a left ideal of R. A submodule of RR 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 }.
<X>
for all xX,
and then it is not difficult to show that
< X > =
xXRx.
M
such that
M=
Rmi.
M such that M=Rm.
M
such that each element mM
can be expressed uniquely as a finite sum
ai
xi,
with a1, . . . ,
anR
and x1, . . . ,
xnX.
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 x1, x2, . . . , xn generate N and y1+N, y2+N, . . . , ym+N generate M/N, then x1, . . . , xn, y1, . . . , ym generate M.
The module RR
is the prototype of a free module, with generating set {1}.
If RM is a module,
and XM,
we say that the set X is
linearly independent if
ai
xi=0 implies
ai=0 for i=1,...,n,
for any distinct x1,
x2, . . . ,
xn
X and any
a1,
a2, . . . ,
an
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(m1 + m2) = f(m1) + f(m2) and f(rm) = rf(m)
for all rR and all m,
m1,
m2M.
The set of all R-homomorphisms from M into N
is denoted by
HomR(M,N) or Hom(RM,RN).
For an R-homomorphism fHomR(M,N)
we define its
kernel
as
ker(f) = { m M | f(m) = 0 }.
EndR(M).
10.1.4 Proposition. Let M be a free left R-module, with basis X. For any left R-module N and any function
:X->N
there exists a unique R-homomorphism f:M->N with
f(x)=(x), for all
xX.
10.1.5 Theorem.
Let N, N0,
M0 be submodules of
RM.
(a)
N0 /
(N0
M0)
(N0 +
M0) /
M0.
(b)
If N0
N, then
(M / N0) /
(N / N0)
M / N.
(c)
If N0
N, then
N
(N0 +
M0) =
N0 +
(N
M0).
10.1.6 Lemma.
Let X be any subset of the module RM.
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 N
M 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
RM
and any submodule N of M with m
N,
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 }
Ann (M) = { r R |
r m = 0 for all m M }.
10.1.10 Definition. A nonzero module RM 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 RM is simple, then EndR(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.
} I
be a collection of left R-modules indexed by the set I.
The
direct product
of the modules
{M} I
is the Cartesian product
IM,
with componentwise addition and scalar multiplication.
That is, if
x,y IM,
with components
x,
yM
for all
I,
then x+y is defined to be the element with components
(x+y)=x+y,
for all
I.
If rR,
then rx is defined to be the element with components
(rx)=rx,
for all
I.
The submodule of
IM
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
IM.
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: IM->M
be the projection defined by
p(m)=m,
for all
m IM,
and let
i:M-> IM
be the inclusion defined for all
xM
by i(x)=m,
where
m=x and
mi=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-> IM
such that
pf=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: IM->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
RI,
for some index set I.
(b)
The module M is a homomorphic image
of a free module.
10.2.4 Proposition.
Let M and M1, . . . ,
Mn be left R-modules.
Then
M
M1
M2
. . .
Mn
if and only if there exist R-homomorphisms
ij:Mj->M
and
pj:M->Mj
for j=1, . . . , n such that
pj
ik =
jk
and
i1p1
+ . . . +
inpn
= 1M.
10.2.5 Proposition.
Let A1,
A2, . . . ,
An be left ideals of the ring R.
(a)
R = A1
A2
. . .
An
if and only if there exists a set
e1,
e2, . . . ,
en
of orthogonal idempotent elements of R such that
Aj=Rej
for
1
jn
and
e1 +
e2 + . . . +
en = 1.
(b)
The left ideals Aj 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=M1
M2
. . .
Mn,
where
Mj is a module over the ring
Aj,
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=1N.
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=1L.
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=1N.
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
RM:
(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 RM 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 RP:
(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 RM is called projective if it is isomorphic to a direct summand of a free module.
M2
M3
. . .
M2
M3
. . .
10.3.2 Definition.
A ring R is said to be
left Noetherian
if the module RR is Noetherian.
A ring R is said to be
left Artinian
if the module RR 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
RM:
(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
RM 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[x1,x2,...,xn] 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 Z, which is not Artinian.