soft question - What are your favorite instructional counterexamples?

Here is a useful example of counter-examples in commutative ring theory;

Let $R=P(mathbb{N})$ be the power set of $mathbb{N}.$ It has a ring structure $(R, +, times)$ where $+$ is the symmetric difference of sets and $times$ is the intersection of sets.

Applications:

Obviously, $R$ is a commutative ring with $1$, ($mathbb{N}$ is the $1$).

1) Let $R$ be a commutative ring with $1$ and a multiplicative closed set of $R$. If $R$ is Noetherian (Artinian) ring then $S^{-1}R$ is Noetherian (Artinian). Does the converse hold?

No, it doesn't.

Using the above example, for any prime ideal $p$ of $R$, $R_p$ (the localization at $p$) is Noetherian (Artinian) while, $R$ is not Noetherian (Artinian).

Outline:

Consider P({1}) $subset$ P({1,2}) $subset...$ and $P(mathbb{N}) supset$ P($mathbb{N} setminus${1}) $supset$ P($mathbb{N} setminus${1,2}) $supset ...$ showing that $R$ is neither Noetherian nor Artinian ring.

It is easy to verify that $R_p$ is isomorphic to $mathbb{Z}/2$, hence it is both Noetherian & Artinian. (Every element of $R_p$ is either $0/1$ or a invertible.)

2) Let $R$ be an integral domain (also commutative with $1$), then for every multiplicative closed set of $R$, $S^{-1}R$ is an integral domain, hence for every $R_p.$ Does the converse hold?

By the above example, it doesn't, since $(P(mathbb{N}),+,times)$ is not an integral domain.

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