Since it is essentially independent of the rest of the book, it can be read at any time. All the necessary stated theoretical prerequisites, including complete proofs of the relevant facts of cardinal arithmetic, are given in the Introduction. But the exercises are not an integral part of the text in that sense.
My colleagues at the University of Washington advised on various parts of the manuscript. Thanks are also due to the students who have used preliminary versions of the manuscript over the past four years. I am pleased to acknowledge the assistance of the secretarial staff at the University of Washington.
Furthermore, the reader should consult the introduction to each chapter for information on the interdt:dependency of the different sections of the chapter. Section VIlLI (Chain Conditions) is used extensively in Chapter IX, but Chapter IX is independent of the rest of Chapter VIII.
PRODUCTS
Instead we assume that the reader is completely familiar with the set Z of integers, the set N. The positive integer c ij' is said to be the greatest common divisor of the integers al,a2,. The proof, which proceeds by induction, can be found in Shockley [5 1, p. l7]. m is an equivalence relation on the set of integers Z, which has exactly m equivalence classes. i) The fact tl' in the congruence module m is an equivalence relation is an easy consequence of the proper definitions.
A choice function for S is a function f from the set of all nonempty subsets from S to S such that f(A) s A for all A =F- 0, A C S. The cardinal number of the set N of natural numbers is usually denoted by � (read "alef-zero"). The set N*, the set Z of integers, and the set Q of rational numbers are countable (exercise 3), but the set R of real numbers is not countable (exercise 9).
In fact, A and B need not be disjoint in the definition of the product a{j (Exercise 4). It is shown to be independent of the axiom of choice and other basic axioms of set theory; see P.
From theorem 1.5 the set of equivalence classes (denoted Q/Z) is an abelian (infinite) group, with addition given by a + b = a + b. By Theorem 1.2 the identity element of any subgroup H is the identity element of G and the inverse of an E H is the inverse a-1 of a in G. Find and prove a condition that will imply that U Hi is a subgroup, so that U Ash = (U Ash). a) The set of all subgroups of a group G, partially ordered according to group theory in.
Let a,b be elements of the group G. Let G be an abelian group containing elements a and b of order m and n re. Show that G contains an element whose order is the least common multiple of m and n. Let G be the multiplicative set of all nonsingular 2 X 2 matrices with rationals. If G is an abelian group, then the set T of all elements of G of finite order is a subgroup of G. Let H be a subgroup of a group G. i) G is the union of the right [fit. left] cosets of H in G are either disjoint or equal. iv).
If CR is the set of distinct right cosets of H in G and £ is the set of distinct left cosets of H in G, then [CRI = j£j. The index of H in G., denoted [G : H], is the cardinal number of the set of distinct rights [resp. Then no left costel of H (except H itself) is also a right costel. every left coset of K is also a right coset of K. The following conditions on a finite group G are equivalent.
Then � is a congruence relation on G if and only if N is a normal subgroup of G and � is a congruence modulo N. Let g be a category whose objects are all groups; hom( A ,B) is the set of all group homomorphisms f : A �B. A is an identity morphism from f to f in � and that h is an equivalence in � if and only if h is an equivalence in e.
G is the internal weak direct product of the family { Ni I i E I } if and only if every non-identity element of G is a unique product ai1ai:l" · · ain with i�,. 3The normal subgroup generated by a set S c F is the intersection of all normal subgroups of F containing S; see Exercise 5.2 A group G is said to be the group defined by the generators x e X and relations w = e (w e Y) provided G � F /N, where F is the free group of X and N the norn1al subgroup of F generated by Y.
1 is the identity element and the product of two reduced words (� 1 ) must essentially be given by juxtaposition. The group defined by the generator b and the relation bm = e (m E N*) is the cyclic group Zm.
CHAPTER I I
TH E STRUCTU RE
OF G ROU PS
Hint: Use the Fundamental Theorem of Arithmetic to be able to. b) The group Q* is free abelian with basis {pjp is prime in Z}. For a complete classification of all countable torsion groups, see I. Let G be a finitely generated abelian group. i). This implies that the only possible refinements of S are obtained by inserting additional copies of each Gi. penalty of S has exactly the same non-trivial factors as S and is therefore equivalent to S. The next lemma is quite technical.
1 shows that under multiplication the elements of a ring R form a semi-group (a monoid if R has an identity). cable and exponentiation are defined in R. The following statement is often useful in calculations. Let R be a commutative ring with identity and prime characteristic p. R given by r � rP is a homomorphism of rings called the Frobenius homo. The underlying set of R"Jl is exactly R and addition in R"1' coincides with addition in R. plication in RoP, denoted o , is defined by a o h = ha. where ba is the product in R. R·'11 is called the opposite ring of R. b) R has an identity if and only if R. Prove that [R : I] is an ideal of R containing I. b) The center of S is not an ideal in S. c) What is the center of the ring of all n X n matrices over a division ring. a).
If P is an ideal in a not necessarily commutative ring R, then the following con. The set consisting of zero and all zero divisors in a commutative ring with identity contains at least one prime ideal. An ideal M � R in a commutative ring R with identity is maximal if and only if for every r E R - M, there exists x E R such that 1R - rx E M. The ring E of even integers contains a maximal ideal M such that Ej M is also not a field. Let S be a multiplicative subset of a commutative ring R with identity and let I be an ideal in R.
Therefore P is prime by Theorem 2. Let R be a commutative ring with identity and P a prime ideal of R. s-•. Since every ideal of a ring with identity is contained in some maximal ideal (Theorem 2.1 8), the unique maximal ideal of a local ring R must contain every ideal of R (except, of course, R itself). Then S* is a multiplicative subset of R and there is a ring isomorphism S* -IR ro.J T-1(s-1R). a) The set E of positive even integers is a multiplicative subset of Z such that E-1(Z) is the field of rational numbers.
If T. is an integral domain such that R C T C F, then F is (isomorphic) to the quotient field T. Let S be a multiplicative subset of the integral domain R such that 0 + S. A polynomial which is the sum of monomials each having degree k, we say that it is homogeneous of degree k. The degree of f in Xk is the degree of f treated as a polynomial in one indefinite xk over the ring R[xh. In general, this is quite difficult, but some facts are easily established:. i) The units in D[x] are exactly constant polynomials that are units in D [see the proof of Corollary 6.4]. ii) If c E D and c is irreducible in D, then the constant polynomial c is irreducible in D[x] [use Theorem 6. iii) Every first degree polynomial whose leading coefficient is unity in D is irreducible. ductible in D[x].
Show that f . there is a factor g of degree at least k that is irreducible in Z[x). a) Let D be an integral domain and c e D.
CHAPTER IV MODU LES
If R is commutative, it is easy to verify that any left R-module A can be given the structure of a right R-module by defining ar = ra for r e R, a € A (commutativity is needed for (iii) ; for a generalization of this idea to arbitrary rings, see Exercise 1 6). If A is a module with additive identity element OA over a ring R with additive identity OR, then it is easy to show that for all r € R, a e A. If R is a ring, any abelian group can be made into an R -module'. with a trivial module structure by defining ra = 0 for all r ER and a e A. Let A and B be modules over a ring R. If R is a subring, an R-module homomorphism is called a linear trans. When the context is clear, R-module homomorphisms are simply called homo. Note that an R-module homomorphism f: A ----+ B is necessarily a homomorphism of additive abelian groups.
Consequently, the same terminology is used: /is an R-module monomorphism [resp. epimorphism , isomorphism] if it is in . surjective, byective] as a map of sets. i) fis an R-module monomorphism if and only if Ker f= 0;. If R is a ring and f: A � B is an R-module homomorphism, then Ker fis is a submodule of A and Im fis a submodule of B. If { Bi I i E /} is a family of submodules of a module A, then a Bi is easily seen as a submodule of A.
If X is a subset of a module A over a ring R, then the intersection of all submodules of A containing X is called the submodule generated by X (or span . by X). If the index set I is finite, the sum of Bt,. lfR has an identity and C is unitary, then C = Ra. iii). Since A is an additive abelian group, B is a normal subgroup and A/B is a well-defined abelian group.
In light of the preceding results, it is not surprising that the various isomorphism theorems for groups (Theorems are valid, mutatis mutandis, for modules. One only has to look at each stage of the proof to see that each subgroup or homo A over a ring R. ii) ifC C B, then B/C is a submodule of AjC, and here is an R-module isomorph.
If R is a ring and B is a submodule of an R-A module, then there is a one-to-one correspondence between the set of all submodules of A containing B and the set of all submodules of A/B given by C � C/B. Therefore, every submodule of A/B is of the form CjB, where C is a submodule of A containing B. II Ai the direct product of the abelian groups A;, and I: Ai the direct sum of.
