Necessary and sufficient criteria for non-trivial derivations to exist? - MathOverflow

Necessary and sufficient criteria for non-trivial derivations to exist?

2010-06-04T19:40:20Z

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.

Answer by Robin Chapman for Necessary and sufficient criteria for non-trivial derivations to exist?

2010-06-04T20:00:45Z

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.

Answer by Steve Huntsman for Necessary and sufficient criteria for non-trivial derivations to exist?

2010-06-04T20:01:38Z

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).

Answer by Hailong Dao for Necessary and sufficient criteria for non-trivial derivations to exist?

2010-06-04T20:15:20Z

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