zbMATH — the first resource for mathematics

On the differential simplicity of polynomial rings. (English) Zbl 1047.16018
The aim of the paper is to present some new examples of rings with a derivation \(d\) having no nontrivial \(d\)-invariant ideals. Let \(\beta\in \mathbb Q[X,Y]\) be an irreducible homogeneous polynomial of degree \(n\geqslant 3\). Then \(\mathbb C[X,Y]\) is \(d\)-simple with respect to the derivation \[ D=\left[\beta (X+Y) +b\right] \dfrac{\partial}{\partial X} +\beta \dfrac{\partial}{\partial Y} \] for any nonzero \(b\in \mathbb Q\). The proof is based on some geometrical considerations.

16W25 Derivations, actions of Lie algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
Full Text: DOI
[1] J. Archer, Derivations on commutative rings and projective modules over skew polynomial rings, PhD thesis, Leeds University, 1981
[2] Carnicer, M.N., The Poincaré problem in the nondicritical case, Ann. of math., 140, 289-294, (1994) · Zbl 0821.32026
[3] Cohn, P.M., On the structure of GL2 of a ring, Inst. hautes études sci. publ. math., 3, 5-53, (1966)
[4] Coutinho, S.C., d-simple rings and simple \(D\)-modules, Math. proc. Cambridge philos. soc., 125, 405-415, (1999) · Zbl 0935.16017
[5] Esteves, E., The castelnuovo – mumford regularity of a variety left invariant by a vector field on projective space, Math. res. lett., 9, 1-15, (2002) · Zbl 1037.14022
[6] Goodearl, K.R.; Warfleld, R.B., An introduction to noncommutative Noetherian rings, () · Zbl 0679.16001
[7] Harris, J., Algebraic geometry: A first course, () · Zbl 0779.14001
[8] Jordan, D.A., Differentiably simple rings with no invertible derivatives, Quart. J. math. Oxford, 32, 417-424, (1981) · Zbl 0471.13014
[9] Neto, A.Lins, Algebraic solutions of polynomial differential equations and foliations dimension two, (), 192-232
[10] Man, Y.-K.; MacCallum, M.A.H., A rational approach to the prelle – singer algorithm, J. symbolic comput., 24, 31-43, (1997) · Zbl 0922.12007
[11] Mattei, J.-F.; Moussu, R., Holonomie et intégrales premières, Ann. sci. école norm. sup. (4), 13, 469-523, (1980) · Zbl 0458.32005
[12] Mumford, D., The red book of varieties and schemes, Lecture notes in math., 1358, (1999), Springer-Verlag
[13] Zakeri, S., Dynamics of singular holomorphic foliations on the complex projective plane, (), 179-233 · Zbl 1193.37066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.