Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 5.4, Problem 5TFE
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Whether the statement, “The set of all integers is well-ordered” is true or false.
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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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- ************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.arrow_forwardI need diagram with solutionsarrow_forwardT. Determine the least common denominator and the domain for the 2x-3 10 problem: + x²+6x+8 x²+x-12 3 2x 2. Add: + Simplify and 5x+10 x²-2x-8 state the domain. 7 3. Add/Subtract: x+2 1 + x+6 2x+2 4 Simplify and state the domain. x+1 4 4. Subtract: - Simplify 3x-3 x²-3x+2 and state the domain. 1 15 3x-5 5. Add/Subtract: + 2 2x-14 x²-7x Simplify and state the domain.arrow_forward
- Q.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.arrow_forwardListen ANALYZING RELATIONSHIPS Describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0 and (c) f(x) <0. y Af -2 1 2 4x a. The function is increasing when and decreasing whenarrow_forwardBy forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1arrow_forwardif a=2 and b=1 1) Calculate 49(B-1)2+7B−1AT+7ATB−1+(AT)2 2)Find a matrix C such that (B − 2C)-1=A 3) Find a non-diagonal matrix E ̸= B such that det(AB) = det(AE)arrow_forwardWrite the equation line shown on the graph in slope, intercept form.arrow_forward1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).arrow_forward1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.arrow_forward1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231arrow_forward1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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