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This question already has an answer here:

I'm looking for an example a ring and a subring with $R \subset S$ such that $R$ has 1 but $S$ does not. Its easy to choose R to be the trivial ring with $0=1$, but are there any more exotic examples of this phenomenon?

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marked as duplicate by user26857 abstract-algebra Feb 15 '16 at 8:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ @user26857 My question asks for exotic examples. I'm looking for more of an assay of what's out there. The ones offered in the linked question are good, but the more the better. I see no reason to close this question as a duplicate. $\endgroup$ – user312108 Feb 15 '16 at 21:29
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Let K be any nontrivial unital ring.

Let $R = \left\{\left(\begin{smallmatrix} a & 0 \\ 0 & 0 \end{smallmatrix}\right) : a \in K \right\}$, and let $S = \left\{\left(\begin{smallmatrix} a & b \\ 0 & 0 \end{smallmatrix}\right) : a,b \in K \right\}$. Note that, S is a rng under the standard operations in $M_2(K)$ whereas R is a ring with identity $\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)$.

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Let $R$ be your favorite ring with $1$, let $T$ be your favorite ring without $1$, and let $S=R\times T$ (identifying $R$ with $R\times\{0\}\subset S$). Your trivial example is just the special case of this when $R$ is the trivial ring.

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