American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
Go
American Journal of Applied Mathematics and Statistics. 2019, 7(2), 75-78
DOI: 10.12691/ajams-7-2-4
Open AccessArticle

Derivations and Integrations on Rings

Michael Gr. Voskoglou1,

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

Pub. Date: February 09, 2019

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)

Creative CommonsThis 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.