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Suppose $k$ is a field,then the quotient ring $k[\epsilon]/\epsilon^2$ is called the ring of dual numbers over $k$. I learn this from Hartshorne. I wonder why it has this name(maybe this question is a bit soft,or senseless). Are there any interesting things about this ring?Would someone be kind enough to say something about it?Thank you very much!

Martin Sleziak
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user14242
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2 Answers2

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Well, one interesting fact about the dual numbers of $\mathbb{R}$: consider its polynomial ring, and specifically identify an object $f(x) = \sum_{i=0}^n a_ix^i , a_i \in \mathbb{R}[\epsilon]/\epsilon^2$. Now evaluating $f(a + b\epsilon), a,b \in \mathbb{R}$ will yield $f(a) + bf'(a)\epsilon$ (hint: binomial theorem) which allows for automatic differentiation and an interesting approach for non-standard analysis.

Working in a more general $k[\epsilon]/\epsilon^2$, since $(a + b\epsilon)(a^{-1} - ba^{-2}\epsilon) = 1,$ we see that for all nonzero $a$, $a + b\epsilon$ is a unit. So our ring of dual numbers over $k$ has a unique maximal ideal $(\epsilon)$ and the ring is local.

On a note more relating to Hartshorne: let $f: X \rightarrow S$ be a morphism of schemes. Using the ring of dual numbers, one can construct the pointed tangent space of $X$ over $S$, but I'm in no means qualified to talk about that.

JakeR
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  • Thank you very much for the explanatin! – user14242 Aug 11 '11 at 15:52
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    @use Since this does not explicitly explain why they are called "dual numbers", I'm quite puzzled why you have accepted it. Doing so may well prevent others from giving the real answer. – Bill Dubuque Aug 11 '11 at 16:34
  • @Bill Dubuque:The above partially answered my question:the interestng properties of the ring. – user14242 Aug 11 '11 at 23:50
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    If I had to guess, I'd say they're called the ring of dual numbers due to the aforementioned relation to pointed tangent spaces. I'd love to hear an actual explanation behind the name, in any case. – JakeR Aug 12 '11 at 11:48
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The "dual" in "ring of dual numbers" is obscure. However, something possibly worth noting is this. The exterior algebra of a finite dimensional vector space over the field k is a self dual (graded) Hopf algebra: that is to say, the multiplication is dual to the comultplication, the unit is dual to the counit, etc. This seems to have been grasped by Grassmann himself though the notation of his time defeated his attempts to formulate this beautiful duality in a coherent fashion. Now note that the ring of dual numbers over k is isomorphic to the exterior algebra over the 1-dimensional vector space k. So it is a self dual Hopf algebra. Could this have a bearing on the terminology? Note also that the Hopf structure illuminates the spectrum: it is "like" a one-dimensional affine group, the single generating vector being "short". A generic tangent.

  • While I think it is a good idea to think of the ring of dual numbers as the exterior algebra of a one dimensional vector space, I now think that the origin of the term "dual" does not lie in the complicated duality of exterior Hopf algebras but rather in the fact that there are two parts to each element, namely the two gradings, 0, and 1. – Steve Selesnick Mar 08 '18 at 17:35