Differential simplicity and dimension of a commutative ring.

*(English)*Zbl 1011.13014Let \(R\) 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 \(>1\).

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 \(>1\).

Reviewer: R.M.Cohn (New Brunswick)