no. 1-2
Algebra
Some remarks on non-commutative principal ideal rings

Présenté par : Michèle Vergne

Carpentier, Sylvain1 ; De Sole, Alberto2 ; Kac, Victor G.3
1 École normale supérieure, 75005 Paris, France
2 Dipartimento di matematica, University of Rome-1, “La Sapienza”, 00185 Roma, Italy
3 Department of Mathematics, M.I.T., Cambridge, MA 02139, USA
Comptes Rendus. Mathématique, Tome 351 (2013) no. 1-2, pp. 5-8.

Nous démontrons quelques résultats algébriques sur lʼanneau des matrices dʼopérateurs différentiels sur un corps différentiel dans le cas général des anneaux principaux non commutatifs. Ces résultats sont utilisés dans la théorie des structures de Poisson non locales.

We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.01.006
Carpentier, Sylvain 1 ; De Sole, Alberto 2 ; Kac, Victor G. 3

1 École normale supérieure, 75005 Paris, France
2 Dipartimento di matematica, University of Rome-1, “La Sapienza”, 00185 Roma, Italy
3 Department of Mathematics, M.I.T., Cambridge, MA 02139, USA 
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Carpentier, Sylvain; De Sole, Alberto; Kac, Victor G. Some remarks on non-commutative principal ideal rings. Comptes Rendus. Mathématique, Tome 351 (2013) no. 1-2, pp. 5-8. doi : 10.1016/j.crma.2013.01.006. http://www.numdam.org/articles/10.1016/j.crma.2013.01.006/

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[2] Carpentier, S.; De Sole, A.; Kac, V.G. Rational matrix pseudodifferential operators, 2012 (preprint) | arXiv

[3] De Sole, A.; Kac, V.G. Non-local Poisson structures and applications to the theory of integrable systems I, 2012 (preprint) | arXiv

[4] De Sole, A.; Kac, V.G. Non-local Poisson structures and applications to the theory of integrable systems II, 2012 (preprint) | arXiv

[5] Dorfman, I.Ya. Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Sci. Theory Appl., Wiley & Sons, New York, 1993

[6] McConnell, J.C.; Robson, J.C. Non-Commutative Noetherian Rings, Grad. Stud. Math., vol. 30, American Mathematical Society, Providence, RI, 2001

[7] Small, L.W.; Stafford, J.T. Regularity of zero divisors, Proc. Lond. Math. Soc. (3), Volume 44 (1982) no. 3, pp. 405-419

[8] J.T. Stafford, private communication, 2012.

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