Let $R$ be a ring with an identity element $1_R$ which is a domain. Let $S$ be a nontrivial subring of $R$ with identity element $1_S$. Prove that $1_R = 1_S$.
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Hint: $1_S$ is an idempotent element of $R$, and it is $\neq 0$ since $S \neq 0$. What are the idempotent elements of a domain?
Martin Brandenburg
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I'm guessing 1R? If so, is it the only idempotent element of R. If that is true then I can see how one can conclude 1R=1S – chappy form Mar 12 '13 at 01:20
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1An integral domain has exactly two idempotent elements. Try to prove it just using the definitions, no ideas are needed. – Martin Brandenburg Mar 12 '13 at 01:22