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Vector fields (and differential operators of higher order) on a real manifold are often defined in terms of $\mathbb{R}$-algebra $C^\infty(M)$. However, it is not clear to me why is the $\mathbb{R}$-algebra structure so important. For example, the proof of locality of derivations does not seem to use the $\mathbb{R}$-linearity at all, but only the additivity property.

I also don't see any troubles with constructing module of Kähler differentials in the context of ring derivations.

What is the essential difference between developing differential calculus on rings and developing it on algebras? I can see that the latter approach automatically gives us rather rich field of constants. Maybe it is important to have ‘enough constants’ in certain situations?

(I asked myself this question almost immediately after the previous one, ‘Modules over algebras’.)

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It's not that you can't do it, but I don't think it has the properties you want. For example, if you're not working over a field you don't have a sensible notion of tangent space, so you can't say that a derivation gives a family of tangent vectors. Perhaps an enlightening example here is to work through what goes wrong with $\mathbb{Z}$. – Qiaochu Yuan Jul 28 '11 at 17:03
@Qiaochu Thanks, I will try it. – Akater Jul 28 '11 at 17:08

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