Modules

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 rR 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 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 _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 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 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, . . . , m_nM such that

M=Rm_i.

In this case, we say that { m₁, m₂, . . . , m_n } 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= a_i x_i, with a₁, . . . , a_nR and x₁, . . . , x_nX.

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, . . . , x_n generate N and y₁+N, y₂+N, . . . , y_m+N generate M/N, then x₁, . . . , x_n, y₁, . . . , y_m 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 a_i x_i=0 implies a_i=0 for i=1,...,n, for any distinct x₁, x₂, . . . , x_n X and any a₁, a₂, . . . , 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(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

Hom_R(M,N) or Hom(_RM,_RN).

For an R-homomorphism fHom_R(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 Hom_R(M,M) are called endomorphisms, and isomorphisms in Hom_R(M,M) are called automorphisms. The set of endomorphisms of _RM will be denoted by

End_R(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 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 _RM 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 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 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.

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 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->_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 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 M₁, . . . , M_n be left R-modules. Then
M M₁ M₂ ^{. . .} 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 i₁p₁ + . . . + i_np_n = 1_M.

10.2.5 Proposition. Let A₁, A₂, . . . , A_n be left ideals of the ring R.
(a) R = A₁ A₂ ^{. . .} A_n if and only if there exists a set e₁, e₂, . . . , e_n of orthogonal idempotent elements of R such that A_j=Re_j for 1jn and e₁ + e₂ + . . . + 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=M₁ M₂ ^{. . .} 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 _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,...,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 Z, which is not Artinian.

Forward | Back | Table of Contents | About this document