Derivations and Integrations on Rings

Michael Gr. Voskoglou^1,

¹Department of Mathematical Sciences, Graduate T. E. I. of Western Greece, Patras, Greece

Cite this paper

Michael Gr. Voskoglou. Derivations and Integrations on Rings. American Journal of Applied Mathematics and Statistics. 2019; 7(2):75-78. doi: 10.12691/AJAMS-7-2-4

Abstract

In this paper properties are studied of the differential ideals of a ring R and of the iterated skew polynomial rings over R defined with respect to a finite set of commuting derivations of R. The concept of the integration of R associated to a given derivation of R is also introduced and some funamental properties of it are studied. This new concept generalizes basic features of the indefinite integrals.

Keywords

derivations, integrations associated to derivations, differential ideals, iterated skew polynomial rings (ISPRs)

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Voskoglou, M. Gr., “Differential simplicity and dimension of a commutative ring”, Rivista Mathematica University of Parma, 6(4), 111-119, 2001.
[2]	Hart, R., “Derivations on regular local rings of finitely generated type”, Journal of London Mathematical Society, 10, 292-294. 1973.
[3]	Voskoglou, M. Gr., “A Study on Smooth Varieties with Differentially Simple Coordinate Rings”, International Journal of Mathematical and Computational Methods, 2, 53-59, 2017.
[4]	Lequain, Y., “Differential simplicity and complete integral closure, Pacific Journal of Mathematics, 36, 741-751, 1971.
[5]	Voskoglou, M. Gr., “A note on the simplicity of skew polynomial rings of derivation type”, Acta Mathematica Universitatis Ostraviensis, 12, 61-64, 2004.
[6]	Cohn, P. M., Free Rings and their Relations, London Mathematical Society Monographs, Academic Press, 1974.
[7]	Ore, O., “Theory of non commutative polynomials”, Annals of Mathematics, 34, 480-508, 1933.
[8]	Voskoglou, M. Gr., “Simple Skew Polynomial Rings”, Publications De L’Institut Mathematique, 37(51), 37-41, 1985.
[9]	Voskoglou, M. Gr., “Extending Derivations and Endomorphisms to Skew Polynomial Rings”, Publications De L’Institut Mathematique, 39(55), 79-82, 1986.
[10]	Kishimoto, K., “On Abelian extensions of rings I”, Mathematics Journal Okayama University, 14, 159-174, 1969-70.
[11]	Majid, S., “What is a Quantum group?”, Notices of the American Mathematical Society, 53, 30-31, 2006.
[12]	Lopez-Permouth, S., “Matrix Representations of Skew Polynomial Rings with Semisimple Coefficient Rings, Contemporary Mathematics, 480, 289-295, 2009.
[13]	Voskoglou, M. Gr., “Derivations and Iterated Skew Polynomial Rings”, Internatinoal Journal of Applied Mathematics and Informatics, 5(2), 82-90, 2011.
[14]	Jordan, D., “Ore extensions and Jacobson rings”, Journal of London Mathematical Society, 10, 281-291, 1975.
[15]	Voskoglou, M. Gr., “Prime ideals of skew polynomial rings”, Rivista Mathematica University of Parma, 4(15), 17-25 , 1989.