Abstract Algebra Answers

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.

Let σ = (a1 a2 ...ak) ∈ Sn be a cycle let τ ∈ Sn.
i) Check that τ σ τ^−1 = (b1 b2 ···bk), where τ (ai) = bi.
ii) Use the above result to compute τ σ τ^−1 where σ and τ are as in part b).

Show that the map f : Z+iZ → Z2, defined by f(a+ib) = (a−b) (mod 2), is an onto ring homomorphism. Describe kef f. Is it a maximal ideal? Justify your answer.

Factorise 10 in two ways in Z[under-rootof -6]. Hence, show that Z[under-rootof -6] is not a UFD.

Factorise 10 in two ways in Z[p