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Chapter 5 Solutions
Elements Of Modern Algebra
- 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_forwardLet S be the set of all 2X2 matrices of the form [x0x0], where x is a real number.Assume that S is a ring with respect to matrix addition and multiplication. Answer the following questions, and give a reason for any negative answers. Is S a commutative ring? Does S have a unity? If so, identify the unity. Is S an integral domain? Is S a field? [Type here][Type here]arrow_forwardGiven that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forward
- 19. Find a specific example of two elements and in a ring such that and .arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)arrow_forward
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forwardExamples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]arrow_forward
- [Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forwardProve that if a is a unit in a ring R with unity, then a is not a zero divisor.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,