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.

Prove the equivalence of the standard definition of equivalence relation on a group and the given alternative definition (of equivalence relation on a group).

Show that the following are equivalent: (a) ~ is an equivalence relation on a group G (b) ~ is reflexive and, for all elements a, b, c of G: if a ~ b and b ~ c, then c ~ a.

Show that a+b is a unit of a commutative ring with identity

Let R be a commutative ring with identity. Let a,b in R. Suppose that a is a unit, and b^2 = 0. Show that a+b is a unit. See attached file for full problem description.

Properties of Elements of a Ring

Give an example of two elements a,b in a ring R such that a(b)=0 but b(a) <> 0. See attached file for full problem description.

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