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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.

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    @rschwieb Yes, I inferred that. I merely wanted to em$p$hasize that such remarks - if ex$p$anded to answers - may help many more of our readers than the few who may be reading these comments, esp. since comments won't appear in search results. As such, generally, if you write something that may be of value to the global community it is better to place it in an answer, than in a comment.2012-08-20

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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.