308 Rings 252 29. Suppose that a and b belong to a commutative ring R with unity. If 30. Suppose that there is an integer n> 1 such that x" x of some ring. If m is a positive integer x for all elements a is a unit of R and b2 0, show that a + b is a unit of R. O for some a, show and a" 0. that a 31. Give an example of ring elements a and b with the properties that ab 0 but ba 0. х, 32. Let n be an integer greater than 1. In a ring in which x" = x for all 0. 0 implies ba show that ab 33. Suppose that R is a ring such that x3 = x for allx in R. Prove that 0 for all x in R бх = a. Prove that a2n a2 34. Suppose that a belongs to a ring and a for all n 1. = a for all a in Z. Do the same 6 35. Find an integer n > 1 such that a" for Zo Show that no such n exists for Z when m is divisible by the 10 т of some prime. square 36. Let m and n be positive integers and letk be the least common mul- tiple of m and n. Show that mZn nZ kZ. 37. Explain why every subgroup of Z, under addition is also a subring of Zg 38. Is Z a subring of Z,? 39. Suppose that R is a ring with unity 1 and a is an element of R such that a2 = 1. Let S Co ara l rE R}. Prove that S is a subring of R. Does S contain 1? 40. Let M,(Z) be the ring of all 2 X 2 matrices over the integers and let R = {, ;]z} 4.bEZ a + b a Prove or disprove that R is a subring a +b b of M2(Z). 41. Let M2(Z) be the ring of all 2 X 2 matrices over the integers and let R S а a a, bE Z. Prove or disprove that R is a subring a -b b of M(Z). 42. Let R a Prove or disprove that R is a subring b of M,(Z). 43. Let R Zze Z and S {(a, b, c) E Rla+ b = c}. Prove or disprove that S is a subring of R. 44. Suppose that there is a positive even integer n such that a" = a for all elements a of some ring. Show that -a a for all a in the ring.