Answer to Question #45563 in Abstract Algebra for SAPNA

Question

a) Let
S =

a 0
0 b

a; b 2 Z

:
i) Check that S is a subring of M2
(R) and it is a commutative ring with identity.
ii) Is S an ideal of M2
(R)? Justify your answer.
iii) Is S an integral domain? Justify your answer.
iv) Find all the units of the ring S.
v) Check whether
I =

a 0
0 b

a; b 2 Z; 2 j a

:
is an ideal of S.
vi) Show that S ' Z Z where the addition and multiplication operations are componentwise
addition and multiplication.
b) Let G = S
4, H = A4
and K = f1; (1 2)(3 4); (1 3)(2 4); (1 4)(2 3)g.
i) Check that H=K = h(1 2 3)Hi
ii) Check that K is normal in H.(Hint: For each h 2 H,h 62 K, check that hK = Kh.)
iii) Check whether (1 2 3 4))H is the inverse of (1 3 4 2)H in the group S
4
=H.