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.
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