Total rings of fractions and Hermite rings

Abstract

In this paper, we show general properties of total rings of fractions and of Hermite rings. We study the relationships between those rings and the finite dimensional K−algebras. A finite dimensional K−algebra is a commutative algebra with unit such that this is finite dimensional as vector space over a field K. We proof that the finite dimensional K−algebras are total rings of fractions and also Hermite rings. In addition, we show that direct product of fields is another example of total ring of fractions and Hermite ring.

Keywords

Localization, direct product of rings, Hermite ring and finite dimensional $K-$algebra

Author Biography

Claudia Granados Pinzón

Doctora en Matemáticas Universidad de Valladolid (España)

C.C. 63514588

Fecha de Nacimiento: 3 de septiembre de 1976

Wilson Olaya León

Wilson Olaya Le´´ón

Doctor en Matemáticas

Profesor Titular Escuela de Matemáticas 

Universidad Industrial de Santander

References

[1] F. Anderson and K. Fuller, Rings and Categories of Modules, Second Edition, Springer-Verlag, New York, 1992.

[2] M. Arapovic, “Characterizations of the 0- dimensional rings”, Glasnik Matematicki, vol. 18, no. 38, pp. 39-46, 1983.

[3] M.F. Atiyah y I.G. Macdonald, Introducción al álgebra conmutativa, Editorial Reverté S. A., Barcelona, 1980.

[4] N. Bourbaki, Commutative algebra, Springer- Verlag, Translated from the french, Reprint of the 1972 edition, 1989.

[5] D. Eisenbud, Commutative Algebra, with a view toward Algebraic Geometry, Springer-Verlag, New York, 1995.

[6] C. Granados Pinzón, Tesis doctoral: Álgebras finitas sobre un cuerpo. La recta proyectiva, Universidad de Valladolid, Valladolid, 2015

[7] C. Granados Pinzón y W. Olaya León, “K−álgebras finitas conmutativas con unidad, Ingeniería y Ciencia”, vol. 12, no. 24, pp. 31-49,
2016.

[8] C. Granados Pinzón, W. Olaya León y S.Pinzón, “Estimación del cardinal del espectro maximal de un producto de cuerpos”, Ciencia En Desarrollo, vol. 9, no. 2, pp. 83-93, 2018.

[9] E. Hartmann, “Planar Circle Geometries: an introduction to Moebius-, Laguerre- and Minkowski-planes”, Darmstadt University of Technology, 2004. Dis- ponible en: http://www.mathematik.tu- darmstadt.de/ ehartmann/circlegeom-introd.pdf

[10] M. Hashimoto, “Equivariant total ring of frac- tions and factoriality of rings generated by semi- invariants”, Communications in Algebra 43, pp. 1524-1562, 2015.


[11] H. Havlicek and K. List, “A three-Dimensional Laguerre geometry and its visualization, In proceedinhs-Dresden Symposium geometry:constructive and kinematic”, Institut für geome- trie TU Dresden, Dresden pp. 122-129, 2003. arXiv:1304.0223v1 [math.AG] 31 Mar 2013.

[12] T.Y. Lam, Serre ́s Problem on Projective Modu- les, Springer Monographs in Mathematics, 2006.

[13] T.Y. Lam, Serre’s conjecture, Lecture notes in mathematics 635, Springer-Verlag, Berlin-New York, 1978.

[14] H.Matsumura,CommutativeRingTheory,Cambridge University Press, Cambridge, 1989.

[15] A. Mnif and I. Yengui, “An algorithm for uni- modular completion over Noetherian rings”, J. Algebra 316, pp. 483-498, 2007.

[16] M. Roitman, “Completing Unimodular Rows to Invertible Matrices”, J. Algebra 49, pp. 206-211, 1977.

[17] I. Yengui, “The Hermite ring conjecture in dimension one”, J. Algebra 320, pp. 437-441, 2008.