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Chapter 5 Solutions
Elements Of Modern Algebra
- Let 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_forward44. 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_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_forward
- 18. If possible, solve .arrow_forwardLet [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].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
- Examples 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_forwardTrue or False Label each of the following statements as either true or false. If one element in a ring R has a multiplicative inverse, then all elements in R must have multiplicative inverses.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning