# Answer to Question #45563 in Abstract Algebra for SAPNA

Question #45563

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.

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.

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