On the differential simplicity of affine rings

Coutinho, S.; Levcovitz, D.

doi:10.1090/S0002-9939-2014-11652-2

On the differential simplicity of affine rings
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by S. C. Coutinho and D. Levcovitz

Proc. Amer. Math. Soc. 142 (2014), 1701-1704

DOI: https://doi.org/10.1090/S0002-9939-2014-11652-2

Published electronically: February 17, 2014

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Abstract:

We prove that every complex regular affine ring is differentially simple relative to a set with only two derivations.

References

J. Archer, Derivations on commutative rings and projective modules over skew polynomial rings, unpublished, 1981, Thesis (Ph.D.)–Leeds University.
Richard E. Block, Differentiably simple algebras, Bull. Amer. Math. Soc. 74 (1968), 1086–1090. MR 231867, DOI 10.1090/S0002-9904-1968-12056-7
Paulo Brumatti, Yves Lequain, and Daniel Levcovitz, Differential simplicity in polynomial rings and algebraic independence of power series, J. London Math. Soc. (2) 68 (2003), no. 3, 615–630. MR 2009440, DOI 10.1112/S0024610703004708
S. C. Coutinho, $d$-simple rings and simple ${\scr D}$-modules, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 405–415. MR 1656789, DOI 10.1017/S0305004198002965
S. C. Coutinho, On the differential simplicity of polynomial rings, J. Algebra 264 (2003), no. 2, 442–468. MR 1981415, DOI 10.1016/S0021-8693(03)00099-1
S. C. Coutinho, Nonholonomic simple $\scr D$-modules from simple derivations, Glasg. Math. J. 49 (2007), no. 1, 11–21. MR 2337862, DOI 10.1017/S0017089507003370
S. C. Coutinho and J. V. Pereira, On the density of algebraic foliations without algebraic invariant sets, J. Reine Angew. Math. 594 (2006), 117–135. MR 2248154, DOI 10.1515/CRELLE.2006.037
Ada Maria De S. Doering, Yves Lequain, and Cydara Ripoll, Differential simplicity and cyclic maximal ideals of the Weyl algebra $A_2(K)$, Glasg. Math. J. 48 (2006), no. 2, 269–274. MR 2256977, DOI 10.1017/S0017089506003053
Laurence R. Harper Jr., On differentiably simple algebras, Trans. Amer. Math. Soc. 100 (1961), 63–72. MR 130250, DOI 10.1090/S0002-9947-1961-0130250-3
R. Hart, Derivations on regular local rings of finitely generated type, J. London Math. Soc. (2) 10 (1975), 292–294. MR 369369, DOI 10.1112/jlms/s2-10.3.292
Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060
Yves Lequain, Daniel Levcovitz, and José Carlos de Souza Junior, $D$-simple rings and principal maximal ideals of the Weyl algebra, Glasg. Math. J. 47 (2005), no. 2, 269–285. MR 2203494, DOI 10.1017/S001708950500248X
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
Andrzej Maciejewski, Jean Moulin-Ollagnier, and Andrzej Nowicki, Simple quadratic derivations in two variables, Comm. Algebra 29 (2001), no. 11, 5095–5113. MR 1856933, DOI 10.1081/AGB-100106804
A. Seidenberg, Differential ideals in rings of finitely generated type, Amer. J. Math. 89 (1967), 22–42. MR 212027, DOI 10.2307/2373093