ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780357671139
Author: Gilbert
Publisher: CENGAGE L
expand_more
expand_more
format_list_bulleted
Question
Chapter 6.2, Problem 23E
To determine
To prove:
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Prove.
Let f : R → R' be a ring homomorphism. If I' is an ideal of R', then f^-1(I') is an ideal of R.
11. Let R and R' be two rings. A mapping f: R→R' is
called an antihomomorphism, if
f(x+y)=f(x) + f(y) and f(xy) = f(y)f(x) x, y € R.
Let f, g be two antihomomorphisms of a ring R into R. Prove that
fg: R R is a homomorphism.
Use First Isomorphism Theorem to prove that R/ZE S'.
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,...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- 11. Show that defined by is not a homomorphism.arrow_forwardShow that if R, R’, and R’’ are rings, and if ø : R → R’ and ψ : R’ → R’’ are homomorphisms, then the composite function ψ ø : R → R’’ is a homomorphism.arrow_forwardConstruct bijective maps(a) from [0, 1) to (0, 2].(b) from R − {3} to R − {1}(c) from Z to Z − {1, 2}.arrow_forward
- Prove that the dual of l' is isometric to 10⁰.arrow_forwardif f:(R. +:) - (R', +'/) be a ring Homomorphism, and R is integral domain, then R' is integral domain if f is Epimorphism True Falsearrow_forward1) Let R and S be rings. Show that the cartesian product of sets R x S = {(r, ) |r e R, s €s} with the addition (r, s) + (r', s') := (r +r', s + s') and the multiplication (r, s) · (r', s') := (r - r', s - s') is a ring.arrow_forward
- 8. Let R denote the ring Z[i]/(1 + 3i). (b) Sho (d) Show that R=Z/10z.arrow_forwardProve that 2/3Z = Z, as rings.arrow_forwardShow how a projective transformation in â: R°/~ → R°/ induces an affine map a if and only if â maps the line at infinity to itself. Prove that â induces a translation a if and only if â fixes each point on the line at infinity.arrow_forward
- Let ƒ: R² → R² be an isomorphism where 2 ƒ([-².]) = [-²] and ƒ ([?]) = [1] Find f ¹ ƒ ([^-+^]) X 0 2 Iarrow_forwardLet A and B be rings. Define addition and multiplication on the Cartesian product A x B by (x,y) + (x', y') = (x + x', y + y') and (x, y)(x', y') = (xx', yy'). Show that the A x B is also a ring. When is A x B commutative? When is it a ring with unity? Deduce from these facts that Z₂ x Z3 is a commutative ring with unity write down its addition and multiplication tables.arrow_forwardABCD is a square Given ABCD cong A’B’ C’ D’, describe a sequence of transformations that maps ABCD to A’ B’C’ D’.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Ring Examples (Abstract Algebra); Author: Socratica;https://www.youtube.com/watch?v=_RTHvweHlhE;License: Standard YouTube License, CC-BY
Definition of a Ring and Examples of Rings; Author: The Math Sorcerer;https://www.youtube.com/watch?v=8yItsdvmy3c;License: Standard YouTube License, CC-BY