Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 5.4, Problem 8E

If x and y are elements of an ordered integral domain D , prove the following inequalities.

a. x 2     − 2 x y     +     y 2     ≥     0

b. x 2     +     y 2     ≥     x y

c. x 2     +     y 2     ≥     − x y

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Chapter 5 Solutions

Elements Of Modern Algebra

Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...Ch. 5.1 - True or False Label each of the following...
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 - Prob. 4ECh. 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 - Prob. 20ECh. 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 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Suppose that a,b, and c are elements of a ring R...Ch. 5.1 - Prob. 28ECh. 5.1 - 29. For a fixed element of a ring , prove that...Ch. 5.1 - Prob. 30ECh. 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 - Prob. 36ECh. 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 - Prob. 45ECh. 5.1 - 46. Let be a set of elements containing the unity,...Ch. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 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 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 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 - Prob. 12ECh. 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 - Prob. 2TFECh. 5.3 - Prob. 3TFECh. 5.3 - Prob. 4TFECh. 5.3 - Prob. 5TFECh. 5.3 - Prove that the multiplication defined 5.24 is a...Ch. 5.3 - Prove that addition is associative in Q.Ch. 5.3 - Prob. 3ECh. 5.3 - Prob. 4ECh. 5.3 - Prob. 5ECh. 5.3 - Prob. 6ECh. 5.3 - 7. Prove that on a given set of rings, the...Ch. 5.3 - Prob. 8ECh. 5.3 - Prob. 9ECh. 5.3 - Since this section presents a method for...Ch. 5.3 - Prob. 11ECh. 5.3 - Prob. 12ECh. 5.3 - Prob. 13ECh. 5.3 - 14. Let be the set of all real numbers of the...Ch. 5.3 - Prob. 15ECh. 5.3 - Prove that any field that contains an intergral...Ch. 5.3 - Prob. 17ECh. 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 - Prob. 5TFECh. 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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