I have to prove that the ring $(\{0\},+,\cdot)$ is a subring of any ring $(R,+,\cdot).$
Let $S = (\{0\},+,\cdot)$ and $R = (R,+,\cdot)$ then $S$ is a subring of $R$ iff $(R,+,\cdot)$ is a ring and $S \subseteq R$ and $S$ is a ring with the same operations.
As we know $S$ has an identity element of $0 \to 0+0=0$. It has an additive inverse of $0 \to 0-0=0$. It is commutative and associative, and it is 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.
Best Answer
Hint $\ $ By the subring test it suffices to verify $\rm\:0-0,\, 0\cdot 0\,\in\, S,\:$ i.e. $\rm \,S\,$ is closed under subtraction and multiplication.