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a. Give an example where a and b are not zero divisors in a ring R , but the sum a + b is a zero divisor. Give an example where a and b are zero divisors in a ring R with a + b ≠ 0 , and a + b is not a zero divisor. Prove that the set of all elements in a ring R that are not zero divisors is closed under multiplication.

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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

Solutions

Chapter
Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 5.2, Problem 19E
Textbook Problem
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a. Give an example where a and b are not zero divisors in a ring R , but the sum a   +   b is a zero divisor.

Give an example where a and b are zero divisors in a ring R with a   +   b     0 , and a   +   b is not a zero divisor.

Prove that the set of all elements in a ring R that are not zero divisors is closed under multiplication.

(a)

To determine

An example where a and b are not zero divisors in a ring R, but the sum a+b is a zero divisor.

Explanation of Solution

Let R=8={[0],[1],[2],[3],[4],[5],[6],[7]} be a ring.

For [1][0] and [3][0] in 8, there doesn’t exist any [a][0] element in 8.

Such that [1][a]=[0] or [3][a]=[0].

[1] and [3] are not zero divisors in a ring 8

(b)

To determine

An example where a and b are zero divisors in a ring R with a+b0, and a+b is not a zero divisor.

(c)

To determine

To prove: The set of all elements in a ring, R that are not zero divisors, is closed under multiplication.

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Elements Of Modern Algebra
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Ch. 5.1 - Exercises Confirm the statements made in Example...Ch. 5.1 - Exercises 2. Decide whether each of the following...Ch. 5.1 - Exercises 3. Let Using addition and...Ch. 5.1 - Exercises Follow the instructions in Exercise 3,...Ch. 5.1 - Exercises 5. Let Define addition and...Ch. 5.1 - Exercises Work exercise 5 using U=a. Exercise5 Let...Ch. 5.1 - Exercises Find all zero divisors in n for the...Ch. 5.1 - Exercises 8. For the given values of , find all...Ch. 5.1 - Exercises Prove Theorem 5.3:A subset S of the ring...Ch. 5.1 - Exercises 10. Prove Theorem 5.4:A subset of the...Ch. 5.1 - Assume R is a ring with unity e. Prove Theorem...Ch. 5.1 - 12. (See Example 4.) Prove the right distributive...Ch. 5.1 - 13. Complete the proof of Theorem by showing that...Ch. 5.1 - Let R be a ring, and let x,y, and z be arbitrary...Ch. 5.1 - 15. Let and be elements of a ring. Prove that...Ch. 5.1 - 16. Suppose that is an abelian group with respect...Ch. 5.1 - If R1 and R2 are subrings of the ring R, prove...Ch. 5.1 - 18. Find subrings and of such that is not a...Ch. 5.1 - 19. Find a specific example of two elements and ...Ch. 5.1 - Find a specific example of two nonzero elements a...Ch. 5.1 - 21. Define a new operation of addition in by ...Ch. 5.1 - 22. Define a new operation of addition in by and...Ch. 5.1 - Let R be a ring with unity and S be the set of all...Ch. 5.1 - Prove that if a is a unit in a ring R with unity,...Ch. 5.1 - 25. (See Exercise 8.) Describe the units of . 8....Ch. 5.1 - 26. (See Example 8.) Describe the units of...Ch. 5.1 - Suppose that a,b, and c are elements of a ring R...Ch. 5.1 - Let R be a ring with no zero divisors. Prove that...Ch. 5.1 - 29. For a fixed element of a ring , prove that...Ch. 5.1 - For a fixed element a of a ring R, prove that the...Ch. 5.1 - Let R be a ring. Prove that the set S={...Ch. 5.1 - 32. Consider the set . a. Construct...Ch. 5.1 - Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8...Ch. 5.1 - The addition table and part of the multiplication...Ch. 5.1 - 35. The addition table and part of the...Ch. 5.1 - 36. Give an example of a zero divisor in the ring...Ch. 5.1 - 37. Let and be elements in a ring. If is a zero...Ch. 5.1 - An element x in a ring is called idempotent if...Ch. 5.1 - 39. (See Exercise 38.) Show that the set of all...Ch. 5.1 - 40. Let be idempotent in a ring with unity....Ch. 5.1 - 41. Decide whether each of the following sets is...Ch. 5.1 - 42. Let . a. Show that is a...Ch. 5.1 - 43. Let . a. Show that is a...Ch. 5.1 - 44. Consider the set of all matrices of the...Ch. 5.1 - 45. Prove the following equalities in an...Ch. 5.1 - 46. Let be a set of elements containing the unity,...Ch. 5.1 - Prove Theorem 5.13 a. Theorem 5.13 Generalized...Ch. 5.1 - 48. Prove Theorem b. Theorem Generalized...Ch. 5.1 - An element a of a ring R is called nilpotent if...Ch. 5.1 - 50. Let and be nilpotent elements that satisfy...Ch. 5.1 - Let R and S be arbitrary rings. In the Cartesian...Ch. 5.1 - 52. (See Exercise 51.) a. Write out the...Ch. 5.1 - Rework exercise 52 with direct sum 24. Exercise 52...Ch. 5.1 - a. Show that S1={ [ 0 ],[ 2 ] } is a subring of 4,...Ch. 5.1 - 55. Rework Exercise 54 with and. Exercise...Ch. 5.1 - Suppose R is a ring in which all elements x are...Ch. 5.2 - True or False Label each of the following...Ch. 5.2 - [Type here] True or False Label each of the...Ch. 5.2 - [Type here] True or False Label each of the...Ch. 5.2 - Label each of the following as either true or...Ch. 5.2 - Confirm the statements made in Example 3 by...Ch. 5.2 - Consider the set ={[0],[2],[4],[6],[8]}10, with...Ch. 5.2 - Consider the set...Ch. 5.2 - [Type here] Examples 5 and 6 of Section 5.1 showed...Ch. 5.2 - Examples 5 and 6 of Section 5.1 showed that P(U)...Ch. 5.2 - [Type here] Examples 5 and 6 of Section 5.1 showed...Ch. 5.2 - [Type here] 7. Let be the set of all ordered pairs...Ch. 5.2 - Let S be the set of all 2X2 matrices of the form...Ch. 5.2 - Work exercise 8 using be the set of all matrices...Ch. 5.2 - Work exercise 8 using S be the set of all matrices...Ch. 5.2 - Let R be the set of all matrices of the form...Ch. 5.2 - Consider the Gaussian integers modulo 3, that is,...Ch. 5.2 - 13. Work Exercise 12 using , the Gaussian integers...Ch. 5.2 - 14. Letbe a commutative ring with unity in which...Ch. 5.2 - [Type here] 15. Give an example of an infinite...Ch. 5.2 - Prove that if a subring R of an integral domain D...Ch. 5.2 - If e is the unity in an integral domain D, prove...Ch. 5.2 - [Type here] 18. Prove that only idempotent...Ch. 5.2 - a. Give an example where a and b are not zero...Ch. 5.2 - 20. Find the multiplicative inverse of the given...Ch. 5.2 - [Type here] 21. Prove that ifand are integral...Ch. 5.2 - Prove that if R and S are fields, then the direct...Ch. 5.2 - [Type here] 23. Let be a Boolean ring with unity....Ch. 5.2 - If a0 in a field F, prove that for every bF the...Ch. 5.2 - Suppose S is a subset of an field F that contains...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - True or False Label each of the following...Ch. 5.3 - Prove that the multiplication defined 5.24 is a...Ch. 5.3 - Prove that addition is associative in Q.Ch. 5.3 - 3. Show that is the additive inverse of in Ch. 5.3 - Prove that addition is commutative in Q.Ch. 5.3 - 5. Prove that multiplication is associative in Ch. 5.3 - 6. Prove the right distributive property in : Ch. 5.3 - 7. Prove that on a given set of rings, the...Ch. 5.3 - Assume that the ring R is isomorphic to the ring...Ch. 5.3 - 9. Let be the ring in Exercise of Section , and...Ch. 5.3 - Since this section presents a method for...Ch. 5.3 - Work Exercise 10 with D=3. Since this section...Ch. 5.3 - Prove that if D is a field to begin with, then the...Ch. 5.3 - 13. Just after the end of the proof of Theorem ,...Ch. 5.3 - 14. Let be the set of all real numbers of the...Ch. 5.3 - Let D be the Gaussian integers, the set of all...Ch. 5.3 - Prove that any field that contains an intergral...Ch. 5.3 - 17. Assume is a ring, and let be the set of all...Ch. 5.3 - 18. Let be the smallest subring of the field of...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - True or False Label each of the following...Ch. 5.4 - Complete the proof of Theorem 5.30 by providing...Ch. 5.4 - 2. Prove the following statements for arbitrary...Ch. 5.4 - Prove the following statements for arbitrary...Ch. 5.4 - Suppose a and b have multiplicative inverses in an...Ch. 5.4 - 5. Prove that the equation has no solution in an...Ch. 5.4 - 6. Prove that if is any element of an ordered...Ch. 5.4 - For an element x of an ordered integral domain D,...Ch. 5.4 - If x and y are elements of an ordered integral...Ch. 5.4 - 9. If denotes the unity element in an integral...Ch. 5.4 - 10. An ordered field is an ordered integral domain...Ch. 5.4 - 11. (See Exercise 10.) According to Definition...Ch. 5.4 - 12. (See Exercise 10 and 11.) If each is...Ch. 5.4 - 13. Prove that if and are rational numbers such...Ch. 5.4 - 14. a. If is an ordered integral domain, prove...Ch. 5.4 - 15. (See Exercise .) If and with and in ,...Ch. 5.4 - If x and y are positive rational numbers, prove...

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