ELEMENTS OF MODERN ALGEBRA8th EditionGilbert Publisher: CENGAGE LISBN: 9780357671139
ELEMENTS OF MODERN ALGEBRA
Solutions for ELEMENTS OF MODERN ALGEBRA
Problem 1TFE:
True or False
Label each of the following statements as either true or false.
1. Every ring is an...Problem 2TFE:
True or False Label each of the following statements as either true or false. Let R be a ring. The...Problem 3TFE:
True or False Label each of the following statements as either true or false. Let R be a ring. Then...Problem 4TFE:
True or False
Label each of the following statements as either true or false.
4. Both , the set of...Problem 5TFE:
True or False Label each of the following statements as either true or false. If one element in a...Problem 6TFE:
True or False Label each of the following statements as either true or false. Let x and y be...Problem 7TFE:
True or False Label each of the following statements as either true or false. Let R be a ring with...Problem 8TFE:
True or False
Label each of the following statements as either true or false.
8. A unity exists in...Problem 9TFE:
True or False Label each of the following statements as either true or false. Any ring with unity...Problem 10TFE:
True or False Label each of the following statements as either true or false. n is a subring of ,...Problem 1E:
Exercises
Confirm the statements made in Example 3 by proving that the following sets are subrings...Problem 2E:
Exercises
2. Decide whether each of the following sets is a ring with respect to the usual...Problem 3E:
Exercises
3. Let Using addition and multiplication as they are defined in Example 5, construct...Problem 5E:
Exercises
5. Let Define addition and multiplication in by and . Decide whether is a ring with...Problem 6E:
Exercises Work exercise 5 using U=a. Exercise5 Let U=a,b. Define addition and multiplication in P(U)...Problem 7E:
Exercises Find all zero divisors in n for the following values of n a. n=6 b. n=8 n=10 d. n=12 n=14...Problem 9E:
Exercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these...Problem 10E:
Exercises
10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these...Problem 11E:
Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the...Problem 12E:
12. (See Example 4.) Prove the right distributive law in:
.
Example 4 For, let denote the...Problem 13E:
13. Complete the proof of Theorem by showing that for any in a ring .
Theorem Zero Product
...Problem 14E:
Let R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11...Problem 16E:
16. Suppose that is an abelian group with respect to addition, with identity element Define a...Problem 21E:
21. Define a new operation of addition in by with a new multiplication in by.
a. Verify that...Problem 22E:
22. Define a new operation of addition in by and a new multiplication in by.
a. Is a...Problem 23E:
Let R be a ring with unity and S be the set of all units in R. a. Prove or disprove that S is a...Problem 27E:
Suppose that a,b, and c are elements of a ring R such that ab=ac. Prove that is a has a...Problem 31E:
Let R be a ring. Prove that the set S={ xRxa=axforallaR } is a subring of R. This subring is called...Problem 32E:
32. Consider the set .
a. Construct addition and multiplication tables for, using the...Problem 33E:
Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition...Problem 34E:
The addition table and part of the multiplication table for the ring R={ a,b,c } are given in Figure...Problem 35E:
35. The addition table and part of the multiplication table for the ring are given
in Figure...Problem 37E:
37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero...Problem 38E:
An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().Problem 39E:
39. (See Exercise 38.) Show that the set of all idempotent elements of a commutative ring is
...Problem 41E:
41. Decide whether each of the following sets is a subring of the ring. If a set is not a
...Problem 44E:
44. Consider the set of all matrices of the form, where and are real
numbers, with the...Problem 46E:
46. Let be a set of elements containing the unity, that satisfy all of the conditions in
...Problem 49E:
An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set...Problem 50E:
50. Let and be nilpotent elements that satisfy the following conditions in a commutative
...Browse All Chapters of This Textbook
Chapter 1.1 - SetsChapter 1.2 - MappingsChapter 1.3 - Properties Of Composite Mappings (optional)Chapter 1.4 - Binary OperationsChapter 1.5 - Permutations And InversesChapter 1.6 - MatricesChapter 1.7 - RelationsChapter 2.1 - Postulates For The Integers (optional)Chapter 2.2 - Mathematical InductionChapter 2.3 - Divisibility
Chapter 2.4 - Prime Factors And Greatest Common DivisorChapter 2.5 - Congruence Of IntegersChapter 2.6 - Congruence ClassesChapter 2.7 - Introduction To Coding Theory (optional)Chapter 2.8 - Introduction To Cryptography (optional)Chapter 3.1 - Definition Of A GroupChapter 3.2 - Properties Of Group ElementsChapter 3.3 - SubgroupsChapter 3.4 - Cyclic GroupsChapter 3.5 - IsomorphismsChapter 3.6 - HomomorphismsChapter 4.1 - Finite Permutation GroupsChapter 4.2 - Cayley’s TheoremChapter 4.3 - Permutation Groups In Science And Art (optional)Chapter 4.4 - Cosets Of A SubgroupChapter 4.5 - Normal SubgroupsChapter 4.6 - Quotient GroupsChapter 4.7 - Direct Sums (optional)Chapter 4.8 - Some Results On Finite Abelian Groups (optional)Chapter 5.1 - Definition Of A RingChapter 5.2 - Integral Domains And FieldsChapter 5.3 - The Field Of Quotients Of An Integral DomainChapter 5.4 - Ordered Integral DomainsChapter 6.1 - Ideals And Quotient RingsChapter 6.2 - Ring HomomorphismsChapter 6.3 - The Characteristic Of A RingChapter 6.4 - Maximal Ideals (optional)Chapter 7.1 - The Field Of Real NumbersChapter 7.2 - Complex Numbers And QuaternionsChapter 7.3 - De Moivre’s Theorem And Roots Of Complex NumbersChapter 8.1 - Polynomials Over A RingChapter 8.2 - Divisibility And Greatest Common DivisorChapter 8.3 - Factorization In F [x]Chapter 8.4 - Zeros Of A PolynomialChapter 8.5 - Solution Of Cubic And Quartic Equations By Formulas (optional)Chapter 8.6 - Algebraic Extensions Of A Field
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More Editions of This Book
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EBK ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780100475755
Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
EBK ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 8220100475757
Elements Of Modern Algebra
8th Edition
ISBN: 9781285965918
Elements of Modern Algebra
5th Edition
ISBN: 9780534373511
Elements of Modern Algebra
6th Edition
ISBN: 9780534402648
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