Let M_2(Z) be the set of all 2╳2 matrices over Z. We know that under the usual matrices addition and multiplication, M_2(Z) is a ring. Let a b S = { ( ) | a, b in Z }, b a be a subset of M_2(Z). Recall that in a ring R an element a in R is an unit if there exists b in R such that ab = ba = 1. (a) Show that S is a subring of M_2(Z). a b (b) Show that { ( ) in M_2(Z) | a^2-b^2 = ±1 } is the b a set of all units of S. (c) Let R = { a + bλ | a,b in Z }, with the property that a + bλ = a' + b'λ in R <=> a = a' and b = b'. We define the addition and multiplication in R by the following: (a + bλ) + (a' + b'λ) = (a + a') + (b + b')λ, (a + bλ)(a' + b'λ) = (aa' + bb') + (ab' + ba')λ. Show that there is a ring isomorphism between R and S. (d) Find the set of all units of R. -- 我好穷啊，我好缺批币啊 ，你有抠抠ㄋㄟ 可怜可怜我吧，施舍一点吧 请到(P)LAY-->(P)AY-->(0)GIVE-->PttFund-->吧 --

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