Show that the sum of two ideals in a ring is also an ideal in that ring.

Let I and J be two ideals in a ring R. Show that I + J = {a + b : a in I and b in J} is an ideal in R.

Ring Unity

Let R be a ring with unity 1 and let S be a subring of R. Is it possible that S has unity e such that e does not equal 1?

Irreducible Polynomials : Eisenstein's Irreducibility Criterion

Let f(x) = x^4 − 8x^3 + 12x^2 − 6x + 2. Show that f(x) is irreducible over the field of rational numbers. I know that it can't be factored... but how do I show its irreducible?

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

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

Prove a subgroup is normal if and only if it is the union of conjugacy classes.

Let G be a group and let H be a subgroup of G. Prove that H is normal if and only if it is the union of conjugacy classes.

Prove properties of a ring with additive identity 0.

Let R be a ring with additive identity 0. Prove the following: (a) For all a in R, a(0) = 0. (b) a(-b)=-(ab). NOTE: see attached word document for clearer notations.