the ring of dual numbers over a field $k$

Question

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!

Answer 1

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.