Abstract Algebra Answers

Mathematical Identities are sometimes indicated by the triple bar ----- when and why ??

Q. Let A and B be two ideals of ring R such that R=A⊕B show that R/A ≅B (R factor A is isomorphic to B) and
R/B ≅A (R factor B is isomorphic to A).

Give counter example to show that Every GCD (Greatest Common Divisor) Domain is not PID(Principal Ideal Domain)?

Q. Let A and B be two ideals of ring R such that R=A⊕B show that R/A ≅B (R factor A is isomorphic to B) and
R/B ≅A (R factor B is isomorphic to A).

what is isomorphism?

how many associative binary operation does set with 3 elements have?

Calculate (2x+3)(3x+4) in Z5[x].

4[8-{5\4}+(5+7)-7]

Which of the following subsets are subgroups of D12? Justify your answer.
i) {x, y, xy, y
2
, y
3
, e} ii) {xy, xy2
, y
2
, e} iii) {x, y
3
, xy3
, e}

Factorise 10 in two ways in Z[
√
−6]. Hence, show that Z[
√
−6] is not a UFD