the ring of dual numbers over a field $k$ 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!
 A: 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.   
A: 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.
