Definition of a module
10.1.1 Definition. Let R be a ring, and let M be an abelian group. Then M
is called a left R-module if there exists a scalar multiplication
 : R � M -> M
denoted by (r,m)=rm,
for all r R and
all m M,
such that for all r,r1, r2
R and all m,
m1, m2
M,
(i) r(m1 + m2) = r m1 + r m2
(ii) ( r1 + r2 ) m = r1 m + r2 m
(iii) r1 ( r2 m ) = ( r1 r2 ) m
(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 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
}.
This is the smallest submodule of M that contains m, so it is called the
cyclic submodule generated by m. More generally, if X is any subset of M,
then the intersection of all submodules of M which contain X is the smallest
submodule of M which contains X. We will use the notation <X> for this
submodule, and call it the submodule generated by X. We must have Rx<X>
for all xX,
and then it is not difficult to show that
< X > =
 xXRx.
10.1.2 Definition. The left R-module M is said to be finitely
generated if there exist m1, m2, . . . , mnM
such that
M= Rmi.
In this case, we say that { m1, m2, . . . , mn
} is a set of generators for M. The module M is called cyclic if
there exists mM
such that M=Rm.
The module M is called a free module if there exists a subset XM
such that each element mM
can be expressed uniquely as a finite sum
m=
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
}.
We say that f is an isomorphism if it is both one-to-one and onto.
Elements of HomR(M,M) are called endomorphisms, and
isomorphisms in HomR(M,M) are called automorphisms. The set of
endomorphisms of RM will be denoted by
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 }
is called the annihilator of m. The ideal
Ann (M) = { r
R | r m = 0
for all m M
}.
is called the annihilator of M.
The module M is called faithful if Ann(M)=(0).
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.
Direct sums and products
10.2.1 Definition. Let {M }
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.
Chain conditiions
10.3.1 Definition. A module R M is said to be Noetherian
if every ascending chain
M1
M2
M3
. . .
of submodules of M must terminate after a finite number of steps.
Similarly, M is said to be Artinian if every descending chain
M1
M2
M3
. . .
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 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.
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