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I'm looking for a simple classification of rings and subrings $S \leq R$ such that $$S \cap R^\times = S^\times,$$ if one exists at all. (Here I assume that both $S$ and $R$ are unital nontrivial rings.)

For example, if $R$ is a field and the subring $S$ satisfies this condition then $S$ must be a subfield of $R$; and if $R$ has a trivial unit group then every subring $S \leq R$ satisfies the relationship.

In case the general classification is too complicated, I'm particularly interested in the case where $R$ is the endomorphism ring (under pointwise addition and composition) of the abelian group (under pointwise addition) of functions $G \to K$ for some set $G$ and abelian group $K$, i.e. $$R = \operatorname{End}(\hat G), \ \ \hat G = (\{f : G \to K\}, \text{pointwise addition}).$$ (So $R^\times$ is the automorphism group of $\hat G$.) Is there a classification of the nontrivial subrings $S$ of this endomorphism ring with the condition that $S \cap R^\times = S^\times$?

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    @RobArthan I'm not saying that _every_ subring of a field is a field -- rather, that if a subring $S$ of a field $R$ satisfies $S \cap R^\times = S^\times$ then $S$ must itself be a subfield of $R$. You're right about the zero ring, though. I keep making little mistakes! – 2017-02-28
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    I've taken the liberty of clarifying the first point in your question. – 2017-02-28

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