Criteria of Subrings Theorem 25.2: Modern Algebra an Introduction
A subset S of a Ring R is a subring of R iff S is nonempty, S is closed under both addition and multiplication of R, and S contains the negative of each of its elements. 
Can anyone prove this? (note it is if and only if)

I'll give you some advice on how to do proofs.
When you want to prove something satisfies a definition, then you need to be familiar with that definition. In this case, you need to know the definition of a subring. From what I vaguely recall, S is a subring if it is also a ring and it has the same multiplicative identity as R (I'm not totally sure on this though).
So how do you start the proof? Well, start with the assumptions in the theorem. In the first direction you have "S is a subring of R" implies $\Rightarrow$ "S is nonempty, S is closed, etc."
This part is easy. HINT: Look at the definition of a subring.
Next, since this is an "if and only if" ($\Leftrightarrow$) statement then you have to also show that it works from the other direction:
S is nonempty, S is closed, S contains negatives" implies $\Rightarrow$ "S is a subring of R"
So work with your assumptions (S is nonempty, S is closed, etc) and figure out how these assumptions alone are enough to show that S satisfies all the other requirements of being a subring (which you will find in the definition of subring).
I would provide more detail but, like I said, I don't remember the definitions of rings or subrings off the top of my head.
