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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!

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up vote 11 down vote accepted

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.

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Thank you very much for the explanatin! – user14242 Aug 11 '11 at 15:52
@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
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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