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

Presented by: Michèle Vergne

Sylvain Carpentier1; Alberto De Sole2; Victor G. Kac3
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, Volume 351 (2013) no. 1-2, pp. 5-8.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.01.006

Sylvain Carpentier 1; Alberto De Sole 2; Victor G. Kac 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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Sylvain Carpentier; Alberto De Sole; Victor G. Kac. Some remarks on non-commutative principal ideal rings. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 5-8. doi : 10.1016/j.crma.2013.01.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.01.006/

[1] S. Carpentier; A. De Sole; V.G. Kac Some algebraic properties of matrix differential operators and their Dieudonné determinant, J. Math. Phys., Volume 53 (2012), p. 063501

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

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

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

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

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

[7] L.W. Small; J.T. Stafford 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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