Differential simplicity and dimension of a commutative ring.(English)Zbl 1011.13014

Let $$R$$ be a commutative ring with identity and with derivation $$d$$. An ideal of $$R$$ closed under $$d$$ is called a $$d$$-ideal. $$R$$ is $$d$$-simple if the only $$d$$-ideals of $$R$$ are the zero ideal and $$R$$ itself. A $$d$$-simple ring $$R$$ contains the field $$F=\{x\in R: dx=0\}$$, so that the characteristic of $$R$$ is either 0 or prime.
It is shown (theorem 2.1) that if the $$d$$-simple ring $$R$$ has prime characteristic $$p$$ then the nilradical of $$R$$ is its only prime ideal. Hence $$R$$ is 0-dimensional. A non-trivial example is given.
The situation for characteristic 0 is much more complicated. A principal result (theorem 2.3) is that if $$R$$ is a complete local ring of characteristic 0 which is $$d$$-simple then the dimension of $$R$$ is 1 (in the terminology used by the author a local ring must be Noetherian). A principal tool in the proof is a theorem of Zariski which permits $$R$$ to be given as a ring of formal power series.
Theorem 2.4 gives an interesting necessary and sufficient condition for a one-dimensional ring $$R=k[y_1, \dots,y_n]$$, $$k$$ a field of characteristic 0, to be $$d$$-simple if $$d$$ is a $$k$$-derivation. An example is given. (In the paragraph immediately preceding this example $$k$$ is used twice where $$R$$ appears to be intended.)
Section 3 shows the existence of polynomial rings in at least two indeterminates, and other related rings, which are $$d$$-simple for appropriate choice of $$d$$ though of dimension $$>1$$.

MSC:

 13N15 Derivations and commutative rings 13C15 Dimension theory, depth, related commutative rings (catenary, etc.)