Show that an annihilator is an ideal of a ring

Let R be a commutative ring and let A be any subset of R. The annihilator of A, denoted by Ann(A), is the set {r in R:r(a)=0 for all a in A}. Show that Ann(A) is an ideal of R. See attached file for full problem description.

Show a subring of a ring of continuous real functions is not an ideal.

Let R be the ring of continuous functions from the reals to the reals. Define A={f in R: f(0) is an even integer}. Show that A is a subring of R, but not an ideal. See attached file for full problem description.

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

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

Ring Theory

If R is a ring and p(x) is included in R[x] then f(x) is the associated polynomial function from R to R. Find a p(x) included in Zmod2[x] such that f(x)=0 for all x included in zmod2. I know that Zmod2 is all the polynomials whose coefficients are 0 and 1 but I have no idea what I am I trying to look for.

Subgroups

Let K and H be subgroups of G. Prove that If H union K is a subgroup of G then either H is a subset of K or K is a subset of H. I understand some lemmas and theorems on intersection of subgroups but I'm unsure about anything related to unions. help please...thank you