I have to prove that the ring $\displaystyle (\{0\},+,\cdot)$ is a subring of any ring $\displaystyle (R,+,\cdot)$

Let S=$\displaystyle (\{0\},+,\cdot)$ and R=$\displaystyle (R,+,\cdot)$ then S is a subring of R iff $\displaystyle (R,+,\cdot)$ is a ring and $\displaystyle S \subseteq R$ 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