I'm reading Serre's "Lie algebra and Lie groups" now, and I found a description of Lie algebra of an algebraic matrix group, given for a free algebra (free as a module) $k'$ with a basis $\lbrace 1, \varepsilon\rbrace$, $\varepsilon^2 = 0$ as all $X \in M(n,k)$ such that $1 + \varepsilon X \in G(k')$, where $G(k')$ is a set of zeros of a number of polinomials.
$k'$ struck me as being extremely close to how smooth infinitesimals are defined in http://arxiv.org/abs/0805.3307 (a link I found on Wiki)
I'm a grad student who recently switched from applied math, so I know very little about algebra, differential geometry etc.
Can you please clarify how is such a free algebra related to smooth infinitesimals, and when were the latter invented, and how close is synthetic differential geometry to this construction? I currently don't know much about synthetic differential geometry, but I know it uses smooth infinitesimals to define differential, tangent space etc. differently from how they are usually defined.
Sorry if the questions are not clear enough :) I want to dive into the interesting subject, I just need a push in the right direction. It amazes me how the example fits the concept of a Lie algebra as being an 'infinitesimal Lie group' when you think of $\varepsilon$ as an infinitesimal, but I don't know what to make out of it, or even if I am right in my intuition.