Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Textbook Question
Chapter 6.1, Problem 29E
Let
a. Prove or disprove that
b. Prove or disprove that
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3. Write a system of linear equations in slope intercept form that has exactly one solution at
the point (3, 4), such that one line has positive slope (but not 1) and the other line has
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Also write your system of equations with both
equations written in standard form with out
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2. Write a system of linear equations in slope-intercept form has exactly one solution at the
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both equations written in standard form
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4. Write a system of linear equations in slope-intercept form that has no solution, such that
(3, 4), and (3,8) are solutions to the first equation, and (0, 4) is a solution to the second
equation.
Also write your system of equations with both
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Chapter 6 Solutions
Elements Of Modern Algebra
Ch. 6.1 - True or False
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - True or false
Label each of the following...Ch. 6.1 - Label each of the following statements as either...Ch. 6.1 - Exercises Let I be a subset of ring R. Prove that...Ch. 6.1 - Prob. 2E
Ch. 6.1 - Prove or disprove each of the following...Ch. 6.1 - Exercises
If and are two ideals of the ring ,...Ch. 6.1 - Prob. 5ECh. 6.1 - Exercises
Find two ideals and of the ring such...Ch. 6.1 - Exercises
Let be an ideal of a ring , and let be...Ch. 6.1 - Exercises
If and are two ideals of the ring ,...Ch. 6.1 - Find the principal ideal (z) of Z such that each...Ch. 6.1 - Let I1 and I2 be ideals of the ring R. Prove that...Ch. 6.1 - Find a principal ideal (z) of such that each of...Ch. 6.1 - 12. Let be a commutative ring with unity. If...Ch. 6.1 - 13. Verify each of the following statements...Ch. 6.1 - 14. Let be an ideal in a ring with unity . Prove...Ch. 6.1 - Let I be an ideal in a ring R with unity. Prove...Ch. 6.1 - Prove that if R is a field, then R has no...Ch. 6.1 - In the ring of integers, prove that every subring...Ch. 6.1 - Let a0 in the ring of integers . Find b such that...Ch. 6.1 - 19. Let and be nonzero integers. Prove that if and...Ch. 6.1 - 20. If and are nonzero integers and is the least...Ch. 6.1 - Prove that every ideal of n is a principal ideal....Ch. 6.1 - 22. Let . Prove .
Ch. 6.1 - 23. Find all distinct principal ideals of for the...Ch. 6.1 - 24. If is a commutative ring and is a fixed...Ch. 6.1 - Given that the set S={[xy0z]|x,y,z} is a ring with...Ch. 6.1 - Prob. 26ECh. 6.1 - Prob. 27ECh. 6.1 - 28. a. Show that the set is a ring with respect to...Ch. 6.1 - 29. Let be the set of Gaussian integers . Let .
...Ch. 6.1 - a. For a fixed element a of a commutative ring R,...Ch. 6.1 - Let R be a commutative ring that does not have a...Ch. 6.1 - 32. a. Let be an ideal of the commutative ring ...Ch. 6.1 - 33. An element of a ring is called nilpotent if...Ch. 6.1 - 34. If is an ideal of prove that the set is an...Ch. 6.1 - Let R be a commutative ring with unity whose only...Ch. 6.1 - 36. Suppose that is a commutative ring with unity...Ch. 6.2 - True or false
Label each of the following...Ch. 6.2 - True or false
Label each of the following...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Label each of the following statements as either...Ch. 6.2 - Each of the following rules determines a mapping...Ch. 6.2 - 2. Prove that is commutative if and only if is...Ch. 6.2 - 3. Prove that has a unity if and only if has a...Ch. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Assume that the set S={[xy0z]|x,y,z} is a ring...Ch. 6.2 - Assume that the set R={[x0y0]|x,y} is a ring with...Ch. 6.2 - 9. For any let denote in and let denote in .
a....Ch. 6.2 - Let :312 be defined by ([x]3)=4[x]12 using the...Ch. 6.2 - 11. Show that defined by is not a homomorphism.
Ch. 6.2 - 12. Consider the mapping defined by . Decide...Ch. 6.2 - Prob. 13ECh. 6.2 -
14. Let be a ring with unity . Verify that the...Ch. 6.2 - In the field of a complex numbers, show that the...Ch. 6.2 - Prob. 16ECh. 6.2 - Define :2()2(2) by ([abcd])=[[a][b][c][d]]. Prove...Ch. 6.2 - Prob. 18ECh. 6.2 - Prob. 19ECh. 6.2 - Prob. 20ECh. 6.2 - Prob. 21ECh. 6.2 - Prob. 22ECh. 6.2 - Prob. 23ECh. 6.2 - Prob. 24ECh. 6.2 - 25. Figure 6.3 gives addition and multiplication...Ch. 6.2 - Prob. 26ECh. 6.2 - 27. For each given value of find all homomorphic...Ch. 6.2 - Prob. 28ECh. 6.2 - 29. Assume that is an epimorphism from to ....Ch. 6.2 - 30. In the ring of integers, let new operations of...Ch. 6.2 - Prob. 31ECh. 6.3 - True or False
Label each of the following...Ch. 6.3 - Prob. 2TFECh. 6.3 - True or False
Label each of the following...Ch. 6.3 - True or False
Label each of the following...Ch. 6.3 - Prob. 5TFECh. 6.3 - Find the characteristic of each of the following...Ch. 6.3 - Find the characteristic of the following rings. 22...Ch. 6.3 - 3. Let be an integral domain with positive...Ch. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - 8. Prove that the characteristic of a field is...Ch. 6.3 - Let D be an integral domain with four elements,...Ch. 6.3 - Let R be a commutative ring with characteristic 2....Ch. 6.3 -
11. a. Give an example of a ring of...Ch. 6.3 - 12. Let be a commutative ring with prime...Ch. 6.3 - Prob. 13ECh. 6.3 - Prob. 14ECh. 6.3 - 15. In a commutative ring of characteristic 2,...Ch. 6.3 - A Boolean ring is a ring in which all elements x...Ch. 6.3 - 17. Suppose is a ring with positive...Ch. 6.3 - Prob. 18ECh. 6.3 - Prob. 19ECh. 6.3 - Let I be the set of all elements of a ring R that...Ch. 6.3 - 21. Prove that if a ring has a finite number of...Ch. 6.3 - 22. Let be a ring with finite number of...Ch. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prove that every ordered integral domain has...Ch. 6.4 - Label each of the following statements as either...Ch. 6.4 - Prob. 2TFECh. 6.4 - According to part a of Example 3 in Section 5.1,...Ch. 6.4 - Let R be as in Exercise 1, and show that the...Ch. 6.4 - Prob. 3ECh. 6.4 - Show that the ideal is a maximal ideal of .
Ch. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Find all maximal ideals of .
Ch. 6.4 - Find all maximal ideals of 18.Ch. 6.4 - Let be the ring of Gaussian integers. Let
...Ch. 6.4 - Let R bethe ring of Gaussian integersas an...Ch. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Prob. 17ECh. 6.4 - Prob. 18ECh. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.4 - Find all prime ideals of .
Ch. 6.4 - Find all prime ideals of .
Ch. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - Prob. 25ECh. 6.4 - . a. Let, and . Show that and are only ideals...Ch. 6.4 - 27. If is a commutative ring with unity, prove...Ch. 6.4 - If R is a finite commutative ring with unity,...
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- Show how you can solve the system of equations by manipulating the algebra tiles while maintaining the balances. On this side of the page, use the addition (elimination) method. Keep track of what you did at each step by writing down the corresponding equivalent equations, as well as what you did to go from one equation to the next. 1. x + 2y = 5 x-2y=1 2. 2x+y=2 x-2y= 6arrow_forwarde) x24 1) Which of these are equivalent to x³? For each expression that is equivalent to x², prove it by using the definition of exponents. For each that is not equivalent to x³, give an example using a specific value for x that shows that it represents a different number. a) (x5) d) f) 10-2 b) (x²) *|*arrow_forwardNow show how you can solve the system of equations by manipulating the algebra tiles while maintaining the balances, using the substitution method. Keep track of what you did at each step by writing down the corresponding equivalent equations, as well as what you did to go from one equation to the next. Δ 1. x + 2y = 5 x-2y=1 2. 2x + y = 2 x-2y= 6arrow_forward
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