Module Theory: An Approach To Linear Algebra - University Of St Andrews

Modules; vector spaces; algebras3If M is an R-module then we shall denote the additive identity of M by 0 M , andthat of R by 0R . The following elementary properties will be used without referencein what follows.Theorem 1.1 Let M be an R-module. Then(1) ( λ R) λ0 M 0 M ;(2) ( x M ) 0R x 0 M ;(3) ( λ R)( x M ) λ( x) (λx) ( λ)x.Moreover, when R is a division ring,(4) λx 0 M implies that λ 0R or x 0 M .Proof (1) We have λ0 M λ(0 M 0 M ) λ0 M λ0 M whence it follows that λ0 M 0M .(2) 0R x (0R 0R )x 0R x 0R x whence 0R x 0 M .(3) By (1), we have 0 M λ0 M λ[x ( x)] λx λ( x) whence λ( x) λx; and, by (2), we have 0 M 0R x [λ ( λ)]x λx ( λ)x whence ( λ)x λx.(4) Suppose now that R is a division ring and that λx 0 M with λ 6 0R . Thenusing the fact that λ has a multiplicative inverse we have x 1R x (λ 1 λ)x λ 1 (λx) λ 1 0 M 0 M . Example 1.1 Every unitary ring R is an R-module; the action R R R is the multiplication in R. Likewise, any field F is an F -vector space.Example 1.2 Every additive abelian group M can be considered as a Z-module;here the action Z M M is given by (m, x) 7 mx where x x · · · x if m 0; {z} mmx 0if m 0; m xif m 0.Example 1.3 The field C of complex numbers can be considered as an R-vectorspace; the action R C C is described by(λ, x i y) 7 λ(x i y) λx iλ y.More generally, if R is a unitary ring and S is a subring of R that contains 1R then Rcan be considered as an S-module; here the action is (s, r) 7 sr.Example 1.4 If R is a unitary ring and n is a positive integer consider the abeliangroup Rn of all n-tuples of elements of R under the component-wise addition(x 1 , . . . , x n ) ( y1 , . . . , yn ) (x 1 y1 , . . . , x n yn ).Define a left action R Rn Rn in the obvious way, namely byr(x 1 , . . . , x n ) (r x 1 , . . . , r x n ).nThen R becomes an R-module. Similarly, if F is a field then F n is an F -vector space.

4Module TheoryExample 1.5 Let R be a unitary ring and let RN denote the set of all mappings f :N R (i.e. the set of all sequences of elements of R). Endow RN with the obviousaddition, namely for f , g RN define f g by the prescription( f g)(n) f (n) g(n).Clearly, RN forms an abelian group under this law of composition. Now define anaction R RN RN by (r, f ) 7 r f where r f RN is given by the prescription(r f )(n) r f (n).This then makes R into an R-module.NEach of the above examples can be made into a right module in the obvious way.Definition 1.2 Let R be a commutative unitary ring. By an R-algebra we shall meanan R-module A together with an internal law of composition A A A, described by(x, y) 7 x y and called multiplication, which is distributive over addition and suchthat( λ R)( x, y A) λ(x y) (λx) y x(λ y).By imposing conditions on the multiplication in the above definition we obtainvarious types of algebra. For example, if the multiplication is associative then A iscalled an associative algebra (note that in this case A is a ring under its internal lawsof addition and multiplication); if the multiplication is commutative then A is calleda commutative algebra; if there is a multiplicative identity element in A then A is saidto be unitary. A unitary associative algebra in which every non-zero element has aninverse is called a division algebra.Example 1.6 C is a division algebra over R.Example 1.7 Let R be a commutative unitary ring and consider the R-module RN ofExample 1.5. Given f , g RN , define the product map f g : N R by the prescriptionnP( f g)(n) f (i)g(n i).i 1It is readily verified that the law of composition described by ( f , g) 7 f g makesRN into an R-algebra. This R-algebra is called the algebra of formal power series withcoefficients in R.The reason for this traditional terminology is as follows. Let t RN be given by§1 if n 1;t(n) 0 otherwise.Then for every positive integer m the m-fold composite maptm t t · · · t{z} mis given byt m (n) §10if n m;otherwise.

Modules; vector spaces; algebras5Consider now (without worrying how to imagine the sum of an infinite number ofelements of RN or even questioning the lack of any notion of convergence) the formalpower series associated with f RN given byPf (i)t i ,ϑ f (0)t 0 f (1)t 1 f (2)t 2 · · · f (m)t m · · · i¾0where t idR , the identity map on R. Since, as is readily seen,0( n N)ϑ(n) f (n),it is often said that f can be represented symbolically by the above formal powerseries.Example 1.8 If R is a unitary ring then the set Matn n (R) of n n matrices over Ris a unitary associative R-algebra.EXERCISES1.1 Let M be an abelian group and let End M be the set of all endomorphisms on M , i.e. theset of all group morphisms f : M M . Show that End M is an abelian group underthe law of composition ( f , g) 7 f g where( x M )( f g)(x) f (x) g(x).Show also that(a) (End M , , ) is a unitary ring;(b) M is an End M -module under the action End M M M given by ( f , m) 7 f · m f (m);(c) if R is a unitary ring and µ : R End M is a ring morphism such that µ(1R ) id M ,then M is an R-module under the action R M M given by (λ, m) 7 λm [µ(λ)](m).1.2 Let R be a unitary ring and M an abelian group. Prove that M is an R-module if andonly if there is a 1-preserving ring morphism f : R End M .[Hint. : For every r R define f r : M M by f r (m) r m. Show that f r End Mand let f be given by r 7 f r . : Use Exercise 1.1(c).]1.3 Let G be a finite abelian group with G m. Show that if n, t Z thenn t (mod m) ( g G) ng t g.Deduce that G is a Z/mZ-module under the action Z/mZ G G which is defined by(n mZ, g) 7 ng. Conclude that every finite abelian group whose order is a prime pcan be regarded as a vector space over a field of p elements.1.4 Let S be a non-empty set and R a unitary ring. If F is the set of all mappings f : S Rsuch that f (s) 0 for almost all s S, i.e. all but a finite number of s S, show that Fis an R-module under the addition defined by ( f g)(s) f (s) g(s) and the actiondefined by (λ f )(s) λ f (s).

6Module Theory1.5 If R is a commutative unitary ring show that the set Pn (R) of all polynomials over Rof degree less than or equal to n is an R-module. Show also that the set P(R) of allpolynomials over R is a unitary associative R-algebra.1.6 If A is a unitary ring define its centre to beCen A {x A ; ( y A) x y y x}.Show that Cen A is a unitary ring. If R is a commutative unitary ring, prove that A isa unitary associative R-algebra if and only if there is a 1-preserving ring morphismϑ : R Cen A.[Hint. : Denoting the action of R on A by (r, a) 7 r · a, define ϑ by ϑ(r) r · 1A. : Define an action by (r, a) 7 r · a ϑ(r)a.]1.7 Let S and R be unitary rings and let f : S R be a 1-preserving ring morphism. If M isan R-module prove that M can be regarded as an S-module under the action S M Mgiven by (s, x) 7 f (s)x.1.8 Show that if V is a vector space over a field F then the set T of linear transformationsf : V V is a unitary associative F -algebra. If F [X ] denotes the ring of polynomialsover F and α is a fixed element of T , show that V can be made into an F [X ]-moduleby the action F [X ] V V defined by(p, x) 7 p ·α x [p(α)](x).

2SUBMODULES; INTERSECTIONS AND SUMSIf S is a non-empty subset of an additive group G then S is said to be a stable subsetof G, or to be closed under the operation of G, if( x, y S)x y S.Equivalently, S is a stable subset of G if the restriction to S S of the law of composition on G induces a law of composition on S, these laws being denoted by the samesymbol without confusion. In this case it is clear that S is a semigroup. By a subgroup of G we mean a non-empty subset that is stable and which is also a group withrespect to the induced law of composition. The reader will recall that a non-emptysubset H of a group G is a subgroup of G if and only if( x, y H)x y H.Definition 2.1 By a submodule of an R-module M we mean a subgroup N of M thatis stable under the action of R on M , in the sense that if x N and λ R thenλx N .It is clear that a non-empty subset N of an R-module M is a submodule of M ifand only if( x, y N )( λ R)x y N and λx N .(2.1)These conditions can be combined into the single condition( x, y N )( λ, µ R)λx µ y N .(2.2)To see this, observe that if (2.1) holds then λx N and µ y N , whence λx µ y λx ( µ y) N . Conversely, if (2.2) holds then taking λ 1R and µ 1R we obtainx y N ; and taking µ 0 we obtain λx N .The notion of a subspace of a vector space is defined similarly. Likewise, we saythat a non-empty subset B of an R-algebra A is a subalgebra of A if( x, y B)( λ R)x y B, x y B, λx B.Example 2.1 Let R be a unitary ring considered as an R-module (Example 1.1). Thesubmodules of R are precisely the left ideals of R. Likewise, if we consider R as aright R-module the its submodules are precisely its right ideals.

8Module Theory Although we agree to omit the adjective ‘left’ when talking about modules, itis essential (except in the case where R is commutative) to retain this adjectivewhen referring to left ideals as submodules of R.Example 2.2 Borrowing some notions from analysis, let C be the set of continuousfunctions f : [a, b] R. Clearly, C can be given the structure of an R-vector space(essentially as in Example 1.5). The subset D that consists of the differentiable functions on [a, b] is then a subspace of C; for, if f , g D then, as is shown in analysis,( λ, µ R) λ f µg D.Example 2.3 If G is an abelian group then the submodules of the Z-module G aresimply the subgroups of G.Example 2.4 The vector space C of Example 2.2 becomes an R-algebra when wedefine a multiplication on C by ( f , g) 7 f g where( x [a, b])( f g)(x) f (x)g(x).It is readily verified that the subspace D is a subalgebra of C.Our first result is a simple but important one.Theorem 2.1 The intersection of any family of submodules of an R-module M is asubmodule of M .ProofT Suppose that (Mi )i I is a family of submodules of M . Then we observe firstthatMi 6 ; since every submodule, being a subgroup, contains the identity elei Iment 0. Now, since each Mi is a submodule, we haveTx, y Mi ( i I) x, y Mii I ( i I) x y MiT x y Mii Iandx TMi , λ R ( i I) λx Mi λx i IConsequently,TMi is a submodule of M .TMi .i I i IThe above result leads to the following observation. Suppose that S is a subset(possibly empty) of an R-module M and consider the collection of all the submodules of M that contain S. By Theorem 2.1, the intersection of this collection is asubmodule of M , and it clearly contains S. It is thus the smallest submodule of M tocontain S. We call this the submodule generated by S and denote it by 〈S〉. We shallnow give an explicit description of this submodule. For this purpose we require thefollowing notion.

Submodules; intersections and sums9Definition 2.2 Let M be an R-module and let S be a non-empty subset of M . Thenx M is a linear combination of elements of S if there exist elements x 1 , . . . , x n in Sand scalars λ1 , . . . , λn in R such thatx nPλ i x i λ1 x 1 · · · λ n x n .i 1We denote the set of all linear combinations of elements of S by LC(S).Theorem 2.2 Let S be a subset of the R-module M . Then〈S〉 §{0}if S ;;LC(S)if S 6 ;.Proof It is clear that if S ; then the smallest submodule that contains S is thesmallest submodule of M , namely the zero submodule {0}. Suppose then that S 6 ;.It is clear that LC(S) is a submodule of M . Moreover, S LC(S) since for every x Swe have x 1R x LC(S). As 〈S〉 is, by definition, the smallest submodule to containS, we therefore have 〈S〉 LC(S). On the other hand, every linear combination ofelements of S clearly belongs to every submodule that contains S and so we havethe reverse inclusion LC(S) 〈S〉, whence the result follows. Definition 2.3 We say that an R-module M is generated by the subset S, or that S isa set of generators of M , when 〈S〉 M . By a finitely generated R-module we meanan R-module which has a finite set of generators.One of the main theorems that we shall eventually establish concerns the structure of finitely generated R-modules where R is a particularly important type of ring(in fact, a principal ideal domain). As we shall see in due course, this structure theorem has far-reaching consequences.Suppose now that (Mi )i I is a family of submodulesof an R-module M and conSsider the submodule of M that is generated by Mi . This is the smallest submodulei Iof M that contains every Mi . By abuse of language it is often referred to as the submodule generated by the family (Mi )i I . It can be characterised in the following way.Theorem 2.3 Let (Mi )i I be a family of submodules of an R-module M . If P? (I)S denotes the set of all non-empty finite subsets of I then the submodule generated by Mii IPconsists of all finite sums of the formm j where J P? (I) and m j M j .j JProof A linear combination of elements ofPm j for some J P? (I).SMi is precisely a sum of the formi I j JBecause of Theorem 2.3, we call the Psubmodule generated by the family (Mi )i Ithe sum of the family and denote it byMi . In the case where the index set I isi I

10Module Theoryfinite, say I {1, . . . , n}, we often writePMi asi InPMi or as M1 · · · Mn . With thisi 1notation we have the following immediate consequences of the above.PCorollary 1 [Commutativity of ]PPMi Mσ(i) .If σ : I I is a bijection thenCorollary 2 [Associativity ofP i Ii I]SIf (I k )k A is a family of non-empty subsets of I with I I k thenk AP P PMi Mi .k A i I ki IP P mi where Jk P? (I k )k J i JkPmi whereand J P? (A). By associativity of addition in M this can be written asi KSK Jk P? (I). Thus the right-hand side is contained in the left-hand side. Ask JPfor the converse inclusion, a typical element of the left-hand side ismi wherei JSJ P? (I). Now J J I (J I k ) so if we define Jk J I k we have Jk P? (I k )k APP P and, by the associativity of addition in M ,mi mi where B P? (A).Proof A typical element of the right-hand side isi Jk B i JkConsequently the left-hand side is contained in the right-hand side.PPCorollary 3 ( i I)Mi Mi M j .i I j6 iProof Take A {1, 2}, I1 {i} and I2 I \ I1 in the above. SPS Note thatMi 6 Mi in general, forMi need not be a submodule. Fori Ii Ii Iexample, take I {1, 2} and let M1 , M2 be the subspaces of the vector spaceR2 given by M1 {(x, 0) ; x R} and M2 {(0, y) ; y R}. We haveM1 M2 R2 whereas M1 M2 R2 .Suppose now that M is an R-module and that A, B are submodules of M . Weknow that A B is the smallest submodule of M that contains both A and B, and thatA B is the largest submodule contained in both A and B. The set of submodules ofM , ordered by set inclusion, is therefore such that every two-element subset {A, B}has a supremum (namely A B) and an infimum (namely A B). Put another way,the set of submodules of M , ordered by set inclusion, forms a lattice. An importantproperty of this lattice is that it is modular, by which we mean the following.Theorem 2.4 [Modular law] If M is an R-module and if A, B, C are submodules ofM with C A thenA (B C) (A B) C.

Submodules; intersections and sums11Proof Since C A we have A C A. Now (A B) C A C and (A B) C B Cand so we have(A B) C (A C) (B C) A (B C).To obtain the reverse inclusion, let a A (B C). Then a A and there exist b B, c C such that a b c. Since C A we have c A and therefore b a c A.Consequently b A B and so a b c (A B) C. EXERCISES2.1 Determine all the subspaces of the R-vector space R2 . Give a geometric interpretationof these subspaces. Do the same for R3 .2.2 Let M be an R-module. If S is a non-empty subset of M , define the annihilator of S inR byAnnR S {λ R ; ( x S) λx 0}.Show that AnnR S is a left ideal of R and that it is a two-sided ideal whenever S is asubmodule of M .2.3 Describe the kernel of the ring morphism µ of Exercise 1.1.2.4 Prove that the ring of endomorphisms of the abelian group Z is isomorphic to the ringZ, and that the ring of endomorphisms of the abelian group Q is isomorphic to the fieldQ.[Hint. Use Exercises 1.1 and 2.3; note that if f End Z then f µ[ f (1)].]2.5 Let M be an R-module. If r, s R show thatr s AnnR M ( x M ) r x sx.Deduce that M can be considered as an R/AnnR M -module. Show that the annihilatorof M in R/AnnR M is zero.2.6 Let R be a commutative unitary ring and let M be an R-module. For every r R let r M {r x ; x M } and M r {x M ; r x 0}. Show that r M and M r are submodules ofM . In the case where R Z and M Z/nZ, suppose that n rs where r and s aremutually prime. Show that r M Ms .[Hint. Use the fact that there exist a, b Z such that r a sb 1.]2.7 Let (Mi )i I be a family of submodules of an R-module M . Suppose that,Sfor every Pfinitesubset J of I, there exists k I such that ( j J) M j Mk . Show that Mi and Mii Ii Icoincide. Show that in particular this arises when I N and the Mi form an ascendingchain M0 M1 M2 · · · .2.8 An R-module M is said to be simple if it has no submodules other than M and {0}.Prove that M is simple if and only if M is generated by every non-zero x M .

12Module Theory2.9 If R is a unitary ring prove that R is a simple R-module if and only if R is a division ring.[Hint. Observe that, for x 6 0, the set Rx {r x ; r R} is a non-zero submodule,whence it must coincide with R and so contains 1R .]2.10 Find subspaces A, B, C of R2 such that(A B) (A C) A (B C).2.11 If M is an R-module and A, B, C are submodules of M such thatA B, A C B C, A C B C,prove that A B.[Hint. A A (A C) · · · ; use the modular law.]2.12 Let V be a vector space over a field F and let α : V V be a linear transformation onV . Consider V as an F [X ]-module under the action defined via α as in Exercise 1.8.Let W be an F [X ]-submodule of V . Prove that W is a subspace of V that satisfies thepropertyx W α(x) W.Conversely, show that every subspace W of V that satisfies this property is an F [X ]submodule of V .

3MORPHISMS; EXACT SEQUENCESThe reader will recall that in the theory of groups, for example, an important partis played by the structure-preserving mappings or morphisms. Precisely, if G and Hare groups whose laws of composition are each denoted by for convenience thena mapping f : G H is called a morphism (or homomorphism) if( x, y G)f (x y) f (x) f ( y).Such a mapping sends G onto a subgroup of H, namely the subgroupIm f { f (x) ; x G}.For such a mapping f we have, with 0G and 0H denoting respectively the identityelements of G and H,(α) f (0G ) 0H ;(β) ( x G) f ( x) f (x).In fact, f (0G ) f (0G 0G ) f

ematics called linear algebra. From the various elementary courses that he has fol-lowed, the reader will recognise this as essentially the study of vector spaces and linear transformations, notions that have applications in several different areas of mathematics. In most elementary introductions to linear algebra the notion of a determinant