Question

Let S = { matrix [ a 0] | a,b ∈ Z } .
[0 b ]

i) Check that S is a subring of M(subscript2)(R) and it is a commutative ring with identity.
ii) Is S an ideal of M(subscript2)(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 = { matrix [ a 0] | a,b ∈ Z, 2 | a } .
[0 b]
is an ideal of S.
vi) Show that S is congruent to Z×Z where the addition and multiplication operations are componentwise addition and multiplication.