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Problem 14 (What Properties Do Subrings Inherit)
Suppose $R$ is a ring, and $S$ is a subring of $R$.
- If $R$ is an integral domain, what properties of being an integral domain does $S$ inherit?
- If $R$ is a field, what properties of being a field does $S$ inherit?
- Challenge: Find an example of a ring $R$ and a subring $S$ so that both $R$ and $S$ have a unity, but the unity of $S$ is not the same as the unity of $R$.
Click to see a hint for the challenge.
You should be able to find the needed example by looking at $\mathbb{Z}_n$ for some values of $n$ less than 10. You won't find the example in any $\mathbb{Z}_p$ for primes $p$, but check the others.
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