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Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to the additive identity.) Part of the motivation for this is that I was thinking about it the other day, and had trouble finding any good example of a semiring with an interesting derivation.

For example, the multiplicative Banach algebra of positive functions is an algebra of the semifield of nonnegative reals. However, the usual definition for derivative breaks down due to the fact that you can have positive functions with negative slope. So, this leads me to wonder if there are any semirings with derivations at all?

As a related question, is there a known classification of all the derivations for an algebra? It feels like this should be a pretty standard thing, but I don't think I've ever encountered it in one of my courses and my initial googling around was not too successful at finding references.

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I assume you mean for all rings (and semirings) to be commutative in this case? – Harry Gindi Jun 4 '10 at 19:50
Yes, that is correct. – Mikola Jun 4 '10 at 20:02
up vote 4 down vote accepted

There is a notion of a universal derivation for an algebra. I'll assume everything is commutative for simiplcitity. If $A$ is a $k$-algebra ($k$ a commutative ring) then there is an $A$-module $\Omega_{A/k}$, the module of Kahler differentials of $A$ over $k$ and a $k$-derivation $d:A\to\Omega_{A/k}$ which is universal for derviations of $A$. That is, if $\delta:A\to M$ is a $k$-derivation from $A$ to an $A$ module, then $\delta=f\circ d$ for a unique $A$-module homomorphism $f$.

There is an explicit desciption of $\Omega_{A/k}$ as $I/I^2$ where $I$ is the ideal in the ring $B=A\otimes_k A$ which is the kernel of the map $\mu:B\to A$ with $\mu(x\otimes y)=xy$. Then $d$ maps $x\in A$ to $1\otimes x-x\otimes 1$. So to find a derivation from $A$ with values in your favourite $A$-module $M$ all one has to do is to find an $A$-homomorphism from $\Omega_{A/k}$ to $M$. Of course $\Omega_{A/k}$ may be hard to determine concretely, and even if that is possible, perhaps it may not be easy to find a homomorphism from that into $M$. Indeed using this method may be no easier than finding a derivation directly :-)

For details see the commutative algebra texts by Eisenbud or Matsumura.

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Thanks! That exactly answers the second question, and combined with the below paper it pretty much settles what I was asking. – Mikola Jun 4 '10 at 20:08

The Banach algebra of bounded functions on a finite set turns out to be semisimple, and therefore carries no nonzero derivations by the results of Johnson, B. E. "Continuity of derivations on commutative algebras". Amer. J. Math. 91, 1 (1969).

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That said, if the set is naturally realized as a "poised" set in $\mathbb{R}^n$, one can use Lagrange interpolation to define "effective" derivations. This is particularly doable in the case of the set of nonnegative multiindices of fixed weight. I would like one day to try to apply this fact to renormalization of lattice gauge theories. – Steve Huntsman Jun 4 '10 at 20:05

For your question in the second paragraph, any semiring can be embedded in one with non-trivial derivation.

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