Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 12, Problem 44E
Suppose that there is a positive even integer n such that
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Chapter 12 Solutions
Contemporary Abstract Algebra
Ch. 12 - The ring {0, 2, 4, 6, 8} under addition and...Ch. 12 - Find an integer n that shows that the rings Zn...Ch. 12 - Show that a ring is commutative if it has the...Ch. 12 - Prove that the intersection of any collection of...Ch. 12 - Let a, b, and c be elements of a commutative ring,...Ch. 12 - Let a andb belong to a ring R and let mbe an...Ch. 12 - Show that if m and n are integers and a and b are...Ch. 12 - Show that a ring that is cyclic under addition is...Ch. 12 - Let R be a ring. The center of R is the set...Ch. 12 - Let R be a commutative ring with unity and let...
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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- 44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.arrow_forward40. Let be idempotent in a ring with unity. Prove is also idempotent.arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4arrow_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forwardAn element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().arrow_forward
- 11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forwardLet R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)arrow_forward
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