5. Prove or disprove: (a) Rand Sare integral domains then Rx Sisan integral domain. (b) If Rand Sare fields, then Rx S i...

Transcribed: Thomas W. Hungerford - Abstrac x b My Questions | bartleby O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf ... Flash Player will no longer be supported after December 2020. Turn off Learn more of 621 + - A) Read aloud V Draw F Highlight O Erase 79 (a) Prove that the set of scalar matrices is a subring of M(R). (b) If Kis a scalar matrix, show that KA = AK for every A in M(R). (c) If K is a matrix in M(R) such that KA = AK for every A in M(R), show that K is a scalar matrix. [Hint: If K = ,let A = Use the fact that KA = AK to show that b = 0 andc = 0. Then make a similar argument with A = to show that a = d.] CourseSuiart 32. Let R be a ring and let Z(R) = {aeR|ar = ra forevery reR}. In other words, Z(R) consists of all elements of R that commute with every other element of R. Prove that Z(R) is a subring of R. Z(R) is called the center of the ring R. [Exercise 31 shows that the center of M(R) is the subring of scalar matrices.] 33. Prove Theorem 3.1. 34. Show that M(Z,) (all 2 × 2 matrices with entries in Z) is a 16-element noncommutative ring with identity. 35. Prove or disprove: (a) If Rand Sare integral domains, then R × S is an integral domain. (b) If Rand Sare fields, then R X S is a field. 36. Let T be the ring in Example 8 and let f, g be given by so if x< 2 if x> 2. if x s2 (2 - x f(x) = {x- 2 if x>2 g(x) = to 3D Show that f, geT and that fg = 07. Therefore Tis not an integral domain. 37. (a) If Ris a ring, show that the ring M(R) of all 2 x 2 matrices with entries in R is a ring. (b) If R has an identity, show that M(R) also has an identity. 38. If R is a ring and aER, let ArR = {rER|ar = 0R}. Prove that AR is a subring of R. AR is called the right annihilator of a. [For an example, see Exercise 16 in which the ring S is the right annihilator of the matrix A.] 39. Let O(V2) = (r + sV2|r, se0}. Show that Q(V2) is a subfield of R. [Hint: To show that the solution of (r +sV2)x= 1 is actually in Q(V2), multiply 1/(r + sV2) by (r – sV2)/(r - sV2).] 40. Let d'be an integer that is not a perfect square. Show that Q(Vd) = {a + bVd|a, beQ} is a subfield of C. [Hint: See Exercise 39.]