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30. 31. 32. 33. 34. 35. 36. 3'7. 38. 39. 40. 41. 42. 43. Suppose that there is an integer n > 1 such thatx\" = x for all elements x of some ring. If m is a positive integer and am = 0 for some a, show that a = 0. Give an example of ring elements a and b with the properties that ab = Obutba #3 0. Let n be an integer greater than 1. In a ring in which A?\" = x for all x, show that ab = 0 implies ba = 0. Suppose that R is a ring such that x3 = x for all x in R. Prove that 6x = 0 for all x in R. 2:"! 2 Suppose that {.1 belongs to a ring and a4 = a2. Prove that a a for all n 2 1. Find an integer n > 1 such that a" = a for all a in 26. Do the same for 210. Show that no such It exists for Zm when m is divisible by the square of some prime. Let m and n be positive integers and let k be the least common mul tiple of m and H. Show that m2 0 112 = 1:2. Explain why every subgroup of Zn under addition is also a subring of Zn. Is 26 a subring of 212? Suppose that R is a ring with unity 1 and a is an element of R such that a2 =1. Let S = {am | r E R}. Prove that S is a subring ofR. Does S contain 1? Let M2(Z) be the ring of all 2 X 2 matrices over the integers and let R = { [ a a + b] a + b b of M2(Z). Let M2(Z) be the ring of all 2 X 2 matrices over the integers and let R = {Lib of 1142(2). wt: 2] of 1142(2). LetR =Z$ZZandS= {(a,b,c)ER|a + b: c}.Proveor disprove that S is a subring of R. a, b E Z } . Prove or disprove that R is a subring a, b E Z } . Prove or disprove thatR is a subring a, b E Z}. Prove or disprove that R is a subring