# Answer to Question #44738 in Abstract Algebra for jasvinder

Question #44738

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.

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