Ring of integers is an integral domain which is not a local ring. Ring of p adic numbers is a local ring which is also an integral domain. Are there any example of a local ring which is not an integral domain? Thank you in advance!
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2What about $10$-adic numbers? – J. W. Tanner May 07 '20 at 22:39
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5$\mathbf Z/p^k\mathbf Z$, where $p$ is prime and $k\ge 2$, is such an example – Bernard May 07 '20 at 22:42
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To add a geometric example to the already mentioned arithmetic ones: $\mathbb C[x]/(x^2) $ – Alex K May 07 '20 at 23:16
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Thank you good examples, I want more detail about ' 10 adic numbers'. If you are Ok, I'd be appreciated if you answer with 10 adic numbers in answer form. – Poitou-Tate May 07 '20 at 23:17
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I'd suggest you read this – J. W. Tanner May 08 '20 at 00:03
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@J.W.Tanner Aren't the $10-$adics the direct sum of $2-$ and $5-$ adics? Then it would certainly not be local. – rschwieb May 08 '20 at 12:41
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For example, $\mathbb Z/4\mathbb Z$.
In general, $R/M^n$ where $M$ is a maximal ideal and $R$ is commutative will be an example, as long as $M\neq \{0\}$.
rschwieb
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This DaRT query returns 10 examples at present, and I think at least half are not of the form I'm describing in this solution. – rschwieb May 08 '20 at 12:32
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@pHotone the n-fold ideal product of M. That is, the set of all finite sums of n-fold products of elements of M. – rschwieb Oct 25 '21 at 03:15