I have to prove that the ring is a subring of any ring

Let S= and R= then S is a subring of R iff is a ring and and S is a ring with the same operations.

As we know S has an identity element of 0 --> 0+0=0

It has an additive inverse of 0 --> 0-0=0

Its commutative and associative

and its distributive over addition.

So my only problem is that I am having difficulties cleaning this up and putting it into a proof.

Maybe you can help me