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There needs to be a consensus throughout the entry on whether a regular local ring needs to be a domain. If not, then certainly it needn't be a UFD (as claimed twice). If yes, then that should be added to the definition in the introduction.
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For the sake of accuracy and efficiency, I think it's better to have just one article on regular rings and regular local rings (as they are essentially the same concept). Since "regular local ring" is a more common term, that means to merge
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That’s not necessarily an argument against the merger, though. By means of merger, the article will acquire a discussion of geometrically regular ring (which may be a good thing from the point of view of this article). The article
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What about the example of say the surface of two cones joined at their pointy ends. A straight line that runs from one to the other is smooth, but the surface is not locally smooth around the point of intersection?!?
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I think the non-example of the dual numbers is outright false. One can easily check the krull dimension to be 1 and the vector space m/m_2 obviously also has. I recommend its removal.
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In the definition of regular, I think a stronger condition than just m=(a_1, \ldots, a_n) is needed. Perhaps that a_1, \ldots a_n is a minimal generating set for m would do.
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A Noetherian Ring R ist a regular ring, if every localization with a prime ideal a regular lokal ring is (see the last two sentences of the article).
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has some examples relevant to this article so that also argues for the merger. But anyway we need a tiebreaker. —-
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Why are all the examples of regular rings in the article completions, rings of formal power series like k
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on
Knowledge (XXG). If you would like to participate, please visit the project page, where you can join
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on 14 September 2020. For the contribution history and old versions of the redirected page, please see
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isn't mentioned on the page (except as a 'see also'), the case is far from clear. Hence
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the ring A is not a lokal ring. We need here the definition
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Oops...I see that it is a consequence of the definition.
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I disagree that it is "more clearly local." I think
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341:{\displaystyle k\!]/(x^{2})}
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267:{\displaystyle k/(x^{2})}
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