Rings: Ideals in Rings

Let S be a subset of a set X. Let R be the ring of real-valued functions on X, and let I be the set of real-valued functions on X whose restriction to S is zero. Show that I is an ideal in R.

Show that a nonzero homomorphism of a simple ring is injective.

Show that a nonzero homomorphism of a simple ring is injective. In particular, a nonzero homomorphism of a field is injective.

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 Polynomial over the field of Rationals

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?