gcd of polynomials

Let f(x) = x^4+2x^3−x^2−4x−2 and g(x) = x^4+x^3−x^2−2x−2. Find the greatest common divisor d(x) of f(x) and g(x) in Q[x]. Find polynomials a(x), b(x) in Q[x] such that d(x) = a(x)f(x) + b(x)g(x).

Family of ideals in a ring

Show that the intersection of any family of ideals in a ring is an ideal. Show that the ideal generated by a subset S of a ring R is the intersection of all ideals J of R such that S <= J <=R.

Prove that an ideal is in a ring

Let R be the ring of 3-by-3 upper triangular matrices and I be the set of upper triangular matrices that are zero on the diagonal. Show that I is an ideal in R.

Show that the ker(@) is an ideal of a ring

Show that if @:R -> S is a ring homomorphism, then the ker(@) is an ideal of R and that @ is injective if. and only if, the kernel is (0).

Equivalence Relations on a Set

Determine all equivalence relations on the set S ={x,y,z}.