Abstract Algebra Answers

if (H,*) is a subgroup of (G,*) and if R is the relation defined on G by aRb iff a * b^-1 is an element of H, then R is an equivalence relation on G, and that aRb iff b is an element of Ha (where Ha={h * a: h is an element of H}), and thus [a] subscript of R=Ha.Let G=S subscript of 4 , and let H=<{(13),(14)}>. Find H, and find all of the equivalence classes of S subscript of 4 under R.

If I=<24,36,42> be an ideal of a ring Z then find a such that I=<a>.

Assume that (G,*) is a group, that g is an element of G and that t is the smallest positive integer such that g^t=e

prove that g^n=e if and only if t |n. use division algorithm and explain why r must = 0

construct cayley table (Z subscript (9) , circle times )and verify its an abelian group.

Assume that a | b times c and that gcf(a,b)=1. Prove that a | c(hint: use the result that gcf(a,b)=1 iff there exist x,y is an element of Z such that a times x+b times y=1)

In the Principal Ideal Domain two non-zero element a,b are coprime if and only if ∃ x,y∈ R such that ax+by=1

Considere the ideal I=<x²-4x+3,x³+3x²-x-3> of the ring Q[x]. Find a polynomial p such that I=<p>.Is Q[x]/I a field? Give reasons for your answer.

Let R be a commutative Ring with identity.let I and J be ideals of R such that I+J=Show that IJ= intersection of I and J.

Give two distinct maximal ideals in the polynomial ring Q[x] with justification.

Prove that R⁵/R² is isomorphic to R³.