Answer to Question #44357 in Abstract Algebra for Jasvinder Singh

Question

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 ' ZZ where the addition and multiplication operations are componentwise
addition and multiplication.