homeranking.info Business ABSTRACT ALGEBRA AN INTRODUCTION HUNGERFORD PDF

# Abstract algebra an introduction hungerford pdf

Buy Abstract Algebra: An Introduction, 3rd Edition on homeranking.info ✓ FREE Thomas W. Hungerford received his M.S. and Ph.D. from the University of Chicago. . other items: algebra 2 textbook pdf, algebra textbook, elementary algebra. Similar searches: Abstract Algebra. An Introduction Hungerford Pdf Solution Manual Algebra-by Thomas W. Hungerford Abstract. Algebra Abstract Algebra Pdf A. Thank you very much for downloading hungerford abstract algebra solutions manual. Maybe Download: Abstract Algebra An Introduction Hungerford Pdf. pdf. Author: SHIRL DEMASTERS Language: English, Spanish, Indonesian Country: Morocco Genre: Health & Fitness Pages: 198 Published (Last): 05.11.2015 ISBN: 601-3-45920-514-1 ePub File Size: 25.82 MB PDF File Size: 20.18 MB Distribution: Free* [*Regsitration Required] Downloads: 21684 Uploaded by: CARIN

Abstract Algebra (An Introduction) - Thomas W. Hungerford - Ebook download as PDF File .pdf), Text File .txt) or read book online. Hungerford, Thomas W. Algebra. Bibliography: p. 1. Algebra I. Ti tie. QAH83 .. ON THE USE OF THIS BOOK. INTRODUCTION. \ v vn. LINEAR. ALGEBRA. An Introduction Hungerford Pdf Solution Manual Algebra-by Thomas W. Hungerford Abstract Algebra. Pdf A First Course In Abstract Algebra Abstract Algebra.

Amazon Inspire Digital Educational Resources. It summarizes the basic rules of logic and the proof techniques that are used throughout this book. Express each of the greatest common divisors a,b in Exercise 1 as a linear combination of a and b. Furthermore, the greatest common divisor of a and b satisfies the inequality. The congruence theme is strongly emphasized in the development of quotient rings. Paul Benedict. The product of this term and the divisor, ix2 g x , is then subtracted from the dividend. Then go back to the beginning and read the proof carefully, line by line. If it says "such and such is true by Theorem 5. If you get stuck, take that part on faith and finish the rest of the proof. Then go back and see if you can figure out the sticky point. When you're really stuck, ask your instructor. He or she will welcome questions that arise from a serious effort on your part.

You can't expect to learn mathematics without doing mathematics, any more than you could learn to swim without getting in the water. That's why there are so many exercises in this book. The exercises in group A are usually straightforward.

If you can't do almost all of them, you don't really understand the material. The exercises in group B often require a reasonable amount of thought - and for most of us, some trial and error as well.

But the vast majority of them are within your grasp. The exercises in group C are usually difficult. Many of the exercises will ask you to prove something. As you build up your skill in reading the proofs of others as discussed above , you will find it easier to make proofs of your own.

The proof techniques presented in Appendix A may also be helpful. Answers or hints for approximately half of the odd-numbered exercises are given at the end of the book. Congruence Congruence of integers may be familiar to you in the form of "clock arithmetic. When you read the statement of a theorem, be sure you know the meaning of all the terms in the statement of the theorem. For example, if it says "every finite integral domain is a field," review the definitions of "integral domain" and "field" - if necessary, use the index to find the definitions.

So it's no wonder that students sometimes have trouble seeing how the various topics tie together, or even if they do. Keeping in mind the three themes mentioned above will help, especially if you. Unlike the usual table of contents, which lists the topics in numerical order by chapter and section, the Thematic Table is arranged in logical order, so you can see how everything fits together.

Arithmetic in Z Revisited 4. Arithmetic in F[x] Division 1. The Division Algorithm 4. Divisibility 4. Divisibility in F[x] f-- Primes and 1. Primes and Unique 4. Polynomial Functions, Roots and Testing Reducibility 4.

Irreducibility in O[x] 4. Congruence in Z and Modular 5. Congruence and Con- 5. Modular Arithmetic 5. The Structure of Zp When p 5. Generally speaking, each block represents a generalization of the block immediately to its left.

Reading from left to right shows how the theme or subtheme listed in the left-hand column is developed first in the integers, then in polynomials, and finally in rings and groups.

Each vertical column shows how all the themes are carried out for the topic listed at the top of the column. Arithmetic in Integral Domains 8.

Unique Factorization Domains - 8. Factorization of Quadratic Integers - 6. Ideals and Quotient Rings 7. Groups 6. Ideals and Congruence 7. Congruence and Normal Subgroups - 6. Quotient Groups 7. Quotient Groups and Homomorphisms - 6. Rings 7. Groups 3. Definition and Examples of Rings 7. Definitions and Examples of Groups 3. Basic Properties of Groups 7.

Subgroups 3. Isomorphism 7. The Structure of Finite Groups 7. The Symmetric Group xxi. Algebra grew out of arithmetic and depends heavily on it. So we begin our study of abstract algebra with a review of those facts from arithmetic that are used frequently in the rest of this book and provide a model for much of the work we do.

We stress primarily the underlying pattern and properties rather than methods of computation. Nevertheless, the fundamental concepts are ones that you have seen before. We also assume the.

If you think of the nonnegative integers laid out on the usual number line. On the other hand, the Well-Ordering Axiom does not hold in the set 'l of all integers there is no smallest negative integer. Nor does it hold. The division process stops when we reach a remainder that is less than the divisor. All the essential facts are contained in the checking procedure, which may be verbally summarized like this:. Here is a formal statement of this idea, in which the dividend is denoted by a, the divisor by b, the quotient by q, and the remainder by r:.

Then there exist unique integers q and r such that. This theorem allows for the possibility that the dividend a may be negative. This is one reason the theorem contains the additional statement that the quotient q and the remainder r are unique. In grade school the idea that a division problem might have more than one correct answer never arises, but without some restrictions on the remainder, such a result is quite possible here.

Simply requiring the remainder do be less than the divisor b does not guarantee a unique quotient or remainder. When the remainder r is also required to be nonnegative as in the theorem , then we do have uniqueness, as will be shown in the proof.

For example, the first step in the long division of by 31 on page 2 amounts to subtracting. Similarly, the second step is subtracting. These subtractions COIltinue until you reach a nonnegative number less than 31 in this case, This number is the remainder, and the number of multiples of 31 namely, that were subtracted is the quotient.

So the division is really this subtraction: In the proof of the theorem, where a is being divided by b, we will subtract multiples of b from a. In other words, we will consider numbers of the form a - bx, where x is any integer in the example, these were the numbers - 31x for various x.

The smallest such nonnegative number is the remainder; the corresponding value of x is the quotient. Proof of Theorem 1. Consider all integers of the form a - bx, where x E 1. We first show that some of these integers must be nonnegative.

There are two possibilities:. Since b is a positive integer, we must have b: Thus we have found integers q and r such that. Suppose, on the contrary, that r: Then r - b: This is a contradiction.

To complete the proof, we must show that q and r are the only numbers with these properties that's what "unique" means in the statement of the theorem. This last equation says that r - rl is an integer multiple of b. Therefore r - rl is one ofOb, 1b, 2b, 3b, etc.

Let n be a positive integer. If you divide 59 by 7 on a calculator, the answer is displayed as 8. Using the Division Algorithm, we see that when 59 is divided by 7, the quotient is 8 and the remainder is 3.

How can you use the calculator to obtain this integer quotient and remainder? More generally, develop an algorithm for use with your calculator that will produce the quotient and remainder for a given dividend and positive divisor. Be sure your algorithm works for both positive and negative dividends. Use the calculator algorithm developed in Exercise 4 to find the quotient and remainder when a is divided by b:.

Prove that the cube of any integer has to be exactly one of these forms: An important case of division occurs when the remainder is 0, that is, when the divisor is a factor of the dividend. Here is a formal definition:. Negative divisors are allowed: An analogous argument shows that every divisor of-a is also a divisor of a.

The common divisors of 12 and 30 are the numbers that divide both 12 and 30, that is, the numbers that appear on both of the preceding lists:. The largest of these common divisors, namely 6, is called the greatest common divisor of 12 and This is an example of the following definition.

The greatest common divisor ged of a and b is the largest integer d that divides both a and b. In other words, d is the ged of a and b provided that. If a and b are not both 0, then their gcd exists and is unique.

The reason is that a nonzero integer has only finitely many divisors, and so there are only a finite number of common divisors. Hence there must be a unique largest one. Furthermore, the greatest common divisor of a and b satisfies the inequality. The only common divisors of 10 and 21 are 1 and Two integers whose greatest common divisor is 1, such as 10 and 21, are said to be relatively prime. Listing all the divisors of two large integers in order to find their gcd can be quite time-consuming.

A relatively quick method for finding gcd' sin such cases is presented in Theorem 1. A little arithmetic shows that something else is true here: For instance,. The following theorem shows that the same thing is possible for any greatest common divisor.

We shall find a particular element of S and show that it is the gcd. Therefore S contains positive integers and hence must contain a smallest positive integer by the Well-Ordering Axiom. Let t denote this smallest positive element of S. To prove this, we first show that t I a. Thus r is a linear combination of a and b, and hence rES. A similar argument shows that t I b. Hence t is a common divisor of a and b. Let c be any other common divisor of a and b, so that c I a and c I b.

The first and last parts of this equation show that cit. Hence c: Thus c: This shows that t is the greatest common divisor d and completes the proof of the theorem. Then d is the greatest common divisor of a and b if and only if d satisfies these conditions:. Then d e: The last paragraph of the proof of Theorem 1. Conversely, suppose d is a positive integer that satisfies the two conditions.

Then d is a common divisor of a and b by i. Ifc is any other common divisor, then c I dby ii. Therefore d is the greatest common divisor. The answer to the following question will be needed on several occasions. If a I be, then under what conditions is it true that a I b or a I c? It is certainly not always true, as this example shows:. Note that 6 has a nontrivial factor in common with S and another in common with 4.

When this is not the case, then there is an affirmative answer to the question:. Consequently, the common divisors of a and b are the same as the common divisors of a and - b. Using similar arguments, we see that. So a method for finding the gcd of two positive integers can also be used to find the gcd of any two integers.

Here is a reasonably efficient method:. Before proving the theorem, we consider a numerical example that should make the process clear. We use the Euclidean Algorithm to find , We now use backsubstitution in the equations above to write 4 as a linear combination of and Thus e I a, so that e is also a common divisor of a and b. Therefore the set S of all common divisors of a and b is the same as the set T of all common divisors of band r. Hence the largest element in S, namely a,b , is the same as the largest element in T, namely b,r.

If b 1 a, then repeated application of Lemma 1. Express each of the greatest common divisors a,b in Exercise 1 as a linear combination of a and b. If aI, a2,. Adapt the proof of Theorem 1. The least common multiple of nonzero integers a and b is the smallest positive integer m such that aim and b 1 m; m is usually denoted [a,b].

Prove that. Prove that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. Integers that have only these four divisors play a crucial role. The integer is prime; to prove this from the definition requires a tedious check of all its possible divisors.

It is not difficult to show that there are infinitely many distinct primes Exercise Because an integer p has the same divisors as - p, we see that. If p and q are both prime and p l q, then 'P must be one of I, - 1, q, - q.

Then p is prime if and only if p has this property:. Proof Suppose p is prime and pi be. Consider the gcd of p and b. Now p,b must be a positive divisor of the prime p. In every case, therefore, p I b or pic. The converse is left to the reader Exercise 4. Proof If pi al a2a If pial' we are finished. If pi a2 a3a If pi a2, we are finished; if not, continue this process, using Theorem 1. After at most n steps, there must be an a, that is divisible by p.

If you factor it "as much as possible," you will find that it is a product of one or more primes. For example,. In this context, we allow the possibility of a "product" with just one factor in case the number we begin with is actually a prime. What was done in these examples can always be done:. Pk is also a product of primes. Consequently we need. Let S be the set of all integers greater than 1 that are not the product of primes.

We shall show that S is empty. Assume on the contrary that S is nonempty. Then S contains a smallest integer m by the Well-Ordering Axiom. Since m E S, m is not itself prime. Since both a and b are less than m the smallest element of S , neither a nor b is in S.

By the definition of S, both a and bare the product of primes, say. We have reached a contradiction: Therefore S must be empty. The only differences are the order of the factors and the insertion of minus signs. You can readily convince yourself that every prime factorization of 45 has exactly three prime factors, say qlq2q3. This is an example of. This prime factorization is unique in the follounng sense: Suppose that n has two prime factorizations, as listed in the statement of the theorem.

By Corollary l. By reordering and relabeling the q's if necessary, we may assume that PI I ql. By Corollary 1. We continue in this manner, repeatedly using Corollary 1. If consideration is restricted to positive integers, then there is a stronger version of unique factorization:. Prove that P is prime if and only if for each a E Prove that P is prime if and only if it has this property: Whenever b and c are integers such that P I be, then P I b or pie.

Prove that p is prime. Prove that n is a perfect square if and only if each rj is even. If n is a positive integer, prove that there exist n consecutive composite integers. Consider the possible remainders when P is divided by 6. Ifn E Use the Fundamental Theorem of Arithmetic or Theorem 1. Use proof by contradiction Appendix A. Prove or disprove: Euclid Prove that there are infinitely many primes. Assume there are only finitely many primes Pl, P2, Copy the proof of Theorem 1.

Does anything have to be changed? The results of the preceding sections are difficult to use on large numbers digits or more. In the past this made no difference. Recently, however, the need for secret codes in the electronic communication of business, military, and scientific information has given primality testing and factoring a new importance.

One of the most promising new coding methods, the RSA cipher system, which is discussed in Chapter 12, makes use of large primes.

Its security depends on the fact that factoring large numbers is very difficult. Even with a computer this may be a formidable task. The following primality test reduces the amount of work considerably. Pk with each Pi prime.

If there are two or more prime factors in this decomposition, then. Theorem 1. List all the numbers frum 2 to n. Let 2,3,5, Now cross out all multiples of 2 on the list. Then cross out all multiples of 3, then all multiples of 5, and so on. Continue until you have crossed out all the multiples of Pk' The numbers remaining on the list are not divisible by any prime less than or equal to Fn. This process is called the Sieve of Erathosthenes.

The efficiency of the preceding methods of finding primes is obviously related to the number of primes less than or equal to n. This number can be approximated by the following theorem, whose proof is beyond the scope of this book. More precisely,. To test an integer n for primality by means of Theorem 1. By Theorem 1. Using this direct approach, it would take more than 63 years to verify that a digit number was prime. Current computer-based primality tests for large numbers depend in part on probabilistic methods.

It can be shown, for example, that every odd prime n has the property that 2n-1 - 1 is divisible by n Lemma Unfortunately, the converse is not true: There are composite numbers n for which n divides 2n-1 - 1. But there are relatively few of them. So if n does divide 2n-1 - 1, then there is a very high probability that n is prime. Although high probability is not the same as certainty, the probabilistic tests can be used to determine likely candidates.

Then other tests can be used to prove primality. At present, the fastest such computer methods can determine if a 1 OO-digit number is prime in 33 seconds. A digit number takes 8 minutes and a digit number one week.

There are a few much larger numbers that have been verified to be prime. They are the Mersenne primes - prime numbers of the form 2P - 1. This last number has 65, digits. After these feats of computation in primality testing, it may come as a bit of a surprise that there are no similar results for factorization. Even if you know that a number is the product of just two large primes, factoring it may be virtually impossible.

The following chart indicates the time needed to find the prime factorization of integers of various sizes, using a state-of-theart computer and the most efficient known algorithm: These times will decrease in coming years as better algorithms and more powerful supercomputers are developed.

Nevertheless, the problem of factoring large numbers will undoubtedly remain formidable for years to come. For example, 3 and 5 are twin primes; so are 11 and Find all pairs of positive twin primes less than A number with some small prime factors can often be factored much more quickly. Prove that every composite three-digit positive integer has a prime factor less than or equal to If n has no prime factor less than or equal to a.

If2P - 1 is prime, prove that P is prime. Prove the contrapositive: If P is composite, so is 2P - 1. The converse is false by Exercise 4 b. Let a, b be nonnegative integers not both 1. Use Exercise 8 above and Exercise 19 in Section 1. Let p, be the nth positive prime in the usual ordering. Use induction to prove that Pn S; 22" This chapter is a bridge between the arithmetic studied in the preceding chapter and the concepts of abstract algebra introduced in the next chapter.

Basic concepts of arithmetic are extended here to include the idea of" congruence modulo n," Congruence leads to the construction of the set Zn of all congruence classes. This construction will serve as a model for many similar constructions in the rest of this book.

It also provides our first example of a system of arithmetic that shares many fundamental properties with ordinary arithmetic and yet differs significantly from it.

## Abstract Algebra (An Introduction) - Thomas W. Hungerford

The concept of "congruence" may be thought of as a generalization of the equality relation. Two integers a and b are equal if their difference is 0 or, equivalently, if their difference is a multiple of o. If n is a positive integer, we say that two integers are congruent modulo n if their difference is a multiple of n. So we have this formal definition:.

The symbol used to denote congruence looks very much like an equal sign. This is no accident since the relation of congruence has many of the same properties as the relation of equality. For example, we know that equality is. The first and last parts of this equation say that n I b - a.

With the equality relation, it's easy to see what numbers are equal to a given number a-just a itself. With congruence, however, the story is different and leads to some interesting consequences.

The congruence class of a modulo n denoted [a] is the set of all those integers that are congruent to a modulo n, that is,. In congruence modulo 3, for instance,.

This ambiguity will not cause any difficulty in what follows because only one modulus will be discussed at a time. Notice,however,that[-1]isthe same class because. Observe that the proof of this theorem does not use the definition of congruence. Instead, it uses only the fact that congruence is reflexive, symmetric, and transitive Theorem 2. Proof of Theorem 2. To do this, let b e [a].

If A and C are two sets, there are usually three possibilities: With congruence classes, however, there are only two possibilities:. Proof If [a] and [c] are disjoint, there is nothing to prove. Suppose that raj n [c] is nonempty. Then there is an integer b with b E raj and b E [c]. Proof We first claim that no two of 0, 1, 2, Then t - s is a positive integer and less than n. Since no two ofO, 1, 2, To complete the proof, we need only show that every congruence class is one of these n classes.

Let a EI. By Theorem 2. There are several points to be careful about here. The elements of In are classes, not single integers. So the statement  E In is true, but the statement 5 E In is not. Furthermore, every element of In can be denoted in many different ways. For example, we know that. Therefore, by Theorem 2. The finite set"ln is closely related to the infinite set"l.

So it is natural to ask if it is possible to define addition and multiplication in "In and do some reasonable kind of arithmetic there. To define addition in "In' we must have some way of taking two classes in "In and producing another class - their sum.

Because addition of integers is defined, the following tentative definition seems worth investigating:. Every element of In can be written in many different ways.

Do we get the same answer if we use  inplaceof and  in place of ? In this case the answer is "yes" because. To get some idea of the kind of thing that might go wrong, consider these five classes of integers:. Every integer is in one of them, and any two of them are either disjoint or identical. Thus you get different answers, depending on which "representatives" you choose from the classes Band C.

Obviously you can't have any meaningful concept of addition if the answer is one thing this time and something else another time. In order to remove the word "tentative" from our definition of addition and multiplication in In, we must first prove that these operations do. Therefore by Theorem 2. Because of Theorem 2. Hla] appears in the left-hand vertical column and Ie] in the top horizontal row of the addition table, for example, then the sum la] 0 Ie] appears at the intersection of the horizontal row containing la] and the vertical column containing [c],.

The key facts about arithmetic in I and the usual titles for these properties are as follows. For all a, h. By using the tables in the example above, you can verify that the first ten of these properties hold in Is and 16 and that property 11 holds in Is and fails. So the proof that properties 1 hold for any 7L,. Proof Properties 1 and 6 are an immediate consequence of the definition of e and 0 in In So the classes of these integers must be the same in By the definition of addition in This proves 2.

The proofs of 3. We have been very careful to distinguish integers in By now, however, you should be reasonably comfortable with the fundamental ideas and familiar with arithmetic in Despite this fact, the new notation is not likely to cause any confusion.

Whenever the context makes clear that we are dealing with 'In, we shall abbreviate the class notation "[a]" and write simply "a" In We shall use an ordinary plus sign for addition in In and either a small dot or juxtaposition for multiplication.

For example, in 'l5 we may write things like. On those few occasions where this usage might cause confusion, we will return to the brackets notation for classes. If a E In and k is a positive integer, then ak denotes the product aaa: Warning Exponents are ordinary integers-not elements of 'In. The set In contains only n elements. To solve an equation in 'In you need only substitute these n elements in the equation to see which ones are solutions.

Solve these equations:. Exercise 5 a may be helpful. See Exercise 5 b. Show that there is no ordering of In such that. Add 1 to each side and get a contradiction.

## Abstract Algebra (An Introduction) - Thomas W. Hungerford

Some of the 1. For instance, the product of nonzero integers in 1. On the other hand, the multiplication table on page 31 shows that the product of nonzero elements in 1.

Indeed, 1. But the multiplication table for 1. The proof of this theorem illustrates the two basic techniques for proving statements that involve 1. Then the properties of congruence and arithmetic in 1. The brackets notation for elements of1. In this case, the brackets notation in 1. Suppose p is prime and [a] oF  in 1.

HenCE p i a by the definition of congruence. Now the gcd of a and p is a positive divisor of p and thus must be either p or 1. Therefore p is prime. According to Theorem 2. When n is not prime, some such equations may have solutions. For example, in It follows from this equation that any common divisor of a and n must divide 1. Without using Theorem 2. You know other mathematical systems, such as the real numbers, in which many of these same rules hold. Your high-school algebra courses dealt with the arithmetic of polynomials.

The fact that similar rules of arithmetic hold in different systems suggests that it might be worthwhile to consider the common features of such systems. In the long run, this might save a lot of work: If we can prove a theorem about one system using only the properties that it has in common with a second system, then the theorem is also valid in the second system.

Results proved for this general theory will apply simultaneously to all the systems covered by the theory. This process of abstraction will allow us to discover the real reasons a particular statement is true or false, for that matter without getting bogged down in nonessential details.

In this way a deeper understanding of all the systems involved should result. So we now begin the development of abstract algebra. This chapter is just the first step and consists primarily of definitions, examples, and terminology. Other names are applied to rings that may have additional properties, as you will see in Section 3. The elementary facts about arithmetic and algebra in arbitrary rings are developed in Section 3. In Section 3. A simple example of this occurs when you write the integers in roman numerals instead of arabic ones. We begin the process of abstracting the common features of familiar systems with this definition:.

These axioms are the bare minimum needed for a system to resemble 7L and 7Ln. But 7L and 7Ln have several additional properties that are worth special mention:. Since the sum or product of two even integers is also even, the closure axioms 1 and 6 hold. Since 0 is an even integer, E has an additive identity element Axiom 4. The other axioms associativity, commutativity, distributivity hold for all integers and therefore are true whenever a, b, c are even.

Consequently E is a commutative ring. Among other things, Axiom 1 fails: The sum of two odd integers is not odd. When a subset 5 of a ring R is itself a ring under the addition and multiplication in R, then we say that 5 is a suhring of R. The zero element of the subring 5 is always the same as the zero element OR of R Exercise 18 in Section 3. Axioms 2, 3, 7, and 8 hold for all elements of a ring R, and so they necessarily hold for the elements of any subset 5.

Consequently, to prove that 5 is a subring, you need only verify that. In these rings, the elements may not be numbers or classes of numbers, and their operations may have nothing to do with "ordinary" addition and multiplication. The element z is the additive identity-the element denoted OR in Axiom 4. It behaves in the same way the number 0 does in I that's why the notation OR is used in the axiom , but z is not the integer 0 - in fact, it's not any kind of number.

Nevertheless, we shall call z the "zero element" of the ring T. In order to verify Axiom. Note that T is not a commutative ring; for instance. Reversing the order of the factors in matrix multiplication may produce a different answer, as is the case here:. So this multiplication is not commutative. With a bit of work, you can verify that M IR is a ring with identity. The zero element is the matrix.

It is proved in calculus that the sum and product of continuous functions are also continuous. You can readily verify that T is a commutative ring with identity. Once again the product of nonzero elements of T may turn out to be the zero element; see Exercise We have seen that some rings do not have the property that the product of two nonzero elements is always nonzero. But some of the rings that do have this property, such as 7l. Note that Axiom 11 is logically equivalent to its contrapositive: If p is prime, then l" is an integral domain by Theorem 2.

You should be familiar with the set 0 of rational numbers, which consists of all fractions alb with a, b E 1. Equality of fractions, addition, and multiplication are given by the usual rules:. It is easy to verify that 0 is an integral domain. But 0 has an additional property that does not hold in Therefore 0 is an example of the next definition. But such equations must be considered in noncommutative rings, where xa may not be the same as ax.

Nevertheless, our primary emphasis will be on fields, and so we begin with examples of them. Since the subset 0 of rational numbers is a field with the same operations, we say 0 is a subfield of R. Equality in C is defined by:. The set C is a field with addition and multiplication given by: In each case the matrix on the right is in K because the entries along the main diagonal upper left to lower right are the same and the entries on the opposite diagonal upper right to lower left are negatives of each other.

Therefore K is closed under addition and multiplication. K is commutative because. It is necessarily infinite: Under the usual addition and multiplication of matrices, H is a noncommutative division ring. See Exercise 31 for details. Whenever the rings in the preceding examples are mentioned, you may assume that addition and multiplication are the operations defined above, unless there is some specific statement to the contrary. You should be aware, however, that a given set such as I may be made into a ring in many different ways by defining different addition and multiplication operations on it.

See Exercises 3 and 17 - 20 for examples. Now that we know a variety of different kinds of rings, we can use them to produce new rings in the following way. Define addition in T by the rule.

The plus sign is being used in three ways here: Since 1. Therefore addition in T is closed. Multiplication is defined similarly:. You can readily verify that Tis a commutative ring with identity. The zero element is 0,0 , and the multiplicative identity is 1,1.

What was done here can be done for any two rings. Define addition and multiplication on the Cartesian product R X S by. Then R X S is a ring. The following subsets of In each case, which axiom fails? Assume associativity and distributivity and show that R is a ring with identity. Is R commutative? Assume associativity and distributivity and show that F is a field. Which ones have an identity?

All matnces of the form b b WIth a, b e R d All matrices of the form: Let T be the ring of all continuous functions from R to IR. Is S a subring ofT? Let Rand S be rings. Let R be a ring and t a fixed element of R. Prove that T is a subring of R. Let E be the set of even integers with ordinary addition. Prove that with these operations E is a commutative ring with identity. Prove that with the new operations e and 0 I is an integral domain.

Prove that with these new operations Q is a commutative ring with identity. Is it an integral domain? Show that R is a subring of C. The addition table and part of the multiplication table for a threeelement ring are given below. Use the distributive laws to complete the multiplication table. Do Exercise 24 for this four-element ring: Show that M 12 all 2 X 2 matrices with entries in 12 is a element noncommutative ring with identity.

Let d be an integer that is not a perfect square. See Exercise Let H be the set of real quaternions and I, i, j, k the matrices defined in the example on page Consider the set lL X lL with the usual coordinatewise addition as in Theorem 3. Show that with these operations lL X lL is a commutative ring with identity. Let S be a set and let P S be the set of all subsets of S.

Define addition and multiplication in P S by the rules. See Exercise 11 for a special case. Consider R X R with the usual coordinatewise addition as in Theorem 3. It is called the center of M R. Exercise 21 is a special case of this result. When you do arithmetic in '1. For instance, subtraction appears regularly, as do cancellation, the various rules for multiplying negative numbers, and the fact that a.

We now show that many of these same properties hold in every ring. Subtraction is not mentioned in the axioms for a ring, and we cannot just assume that such an operation exists in an arbitrary ring. If we want to define a subtraction operation in a ring, we must do so in terms of addition, multiplication, and the ring axioms. The first step is. We can now define negatives and subtraction in any ring by copying what happens in familiar rings such as ' By Theorem 3.

Using notation adapted from '1. In familiar rings, this definition coincides with the known concept of the negative of an element. More important, it provides a meaning for "negative" in any ring. In 7L and other familiar rings, this is just ordinary subtraction.

In other rings we have a new operation:. In junior high school you learned many computational and algebraic rules for dealing with negatives and subtraction. The next two theorems show that these same rules are valid in any ring. Although these facts are not particularly interesting in themselves, it is essential to establish their validity so that we may do arithmetic in arbitrary rings.

Applying Theorem 3. We now carry these concepts over to arbitrary rings. Be careful here. Having established the basic facts about arithmetic in rings, we turn now to algebra and the solution of equations.

But there is one case where such equations must have solutions. We need some terminology to describe this situation. In this case the element u is called the multiplicative inverse of a and is denoted a: Add all three to Cart Add all three to List. These items are shipped from and sold by different sellers. Show details. Buy the selected items together This item: Abstract Algebra: An Introduction, 3rd Edition by Thomas W. FREE Shipping. Ships from and sold by Amazon.

Customers who bought this item also bought. Page 1 of 1 Start over Page 1 of 1. Understanding Analysis Undergraduate Texts in Mathematics. Stephen Abbott. Journey through Genius: The Great Theorems of Mathematics. Jay L. Vector Calculus. Elementary Analysis: Kenneth A. Linear Algebra and Its Applications 5th Edition. About the Author Thomas W. Read more. Product details Hardcover: Cengage Learning; 3 edition July 27, Language: English ISBN Try the Kindle edition and experience these great reading features: Share your thoughts with other customers.

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Hardcover Verified Purchase. I used this book for an undergraduate course on Abstract Algebra. The approach taken by the author is very straightforward, both in its presentation of proofs and its examples. Unlike some books, it is very possible to teach yourself the subject using this book.

It also provides a large number of exercises ranging in difficulty for each section and ample answers in the back. Out of all the books I used in undergrad, this one was one of my favorites because it was easy to use for reference and for independent learning.

I found this book to have several examples in the sections that aided in the solutions to the assignment questions. The treasure of this book is the increasing rigor of the questions.

A teacher is able to assign harder questions to those who excel and are up for the challenge or easier questions for the average student not that any of them are too easy given the subject. Great foundation establishing book to start building in the domain of Abstract Algebra. Paperback Verified Purchase. Kindle Edition Verified Purchase.

I'm not sure why everyone is rating this book so well when it is missing chapters 3. This book is on the defective list for a publishing error and is known to be missing these pages. Per Cengage they are aware the first "few runs" are defective and have stated that these books are not to be sold The publishing company has stated they cannot grant access to the ebook version unless it was purchased through cengagebrain.

There is also no application, simply proof after proof, most that are written poorly, relying on what you're proving to complete the proof. Most of the problems are further proofs that you will need to do in order to accomplish other practice problems.

Definitely not worth the money. Rent it if you must, but get the ebook so you have all the chapters. This also states I bought a Kindle copy, but I have the hardback, so please know that I am talking about that actual hardback book.

This is a great book for those who are learning Abstract Algebra for the first time.