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I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible).
Does it even exist?
Thank you in advance.

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$\mathbb{Z}\times\mathbb{Z}$. There are only $(-1,1),(-1,-1),(1,1),(1,-1)$ as invertible elements but infinitely many zero divisors $(a,0)$ for every $a$.

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The power set of $\: \{\hspace{-0.02 in}0,\hspace{-0.05 in}1\hspace{-0.02 in},\hspace{-0.04 in}2,\hspace{-0.04 in}3,...\hspace{-0.04 in}\} \:$ is an example of such a ring.

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Take $\mathbb{Z}\times R$ for $R$ any nontrivial ring with only finitely many invertible elements.

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  • $\begingroup$ Hey Noah good to see your around. How goes the diss? $\endgroup$ – Robert Lewis Jun 30 '15 at 3:21

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