SciELO - Scientific Electronic Library Online

vol.47 issue2Description of the Nikolski\u\i-Besov Class Decreasing its Smoothness Parameter using Weak Fractional DerivativesThe Stekloff Problem for Rotationally Invariant Metrics on the Ball author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand



Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Revista Colombiana de Matemáticas

Print version ISSN 0034-7426

Rev.colomb.mat. vol.47 no.2 Bogotá July/Dec. 2013


On the Infinitude of Prime Elements

Acerca de la infinitud de elementos primos


1University of Puerto Rico at Mayagüez, Mayagüez, PR, USA. Email:
2Valdosta State University, Valdosta, GA, USA. Email:


Let R be an infinite unique factorization domain with at most finitely many units. We discuss the infinitude of prime elements in R when R is arbitrary and when R satisfies the following property: if f and g are polynomials with coefficients in R such that f(r) divides g(r) for all rε R with f(r)≠ 0, then either g=0 or deg(f) ≤ deg(g).

Key words: Unique factorization domains, Prime elements.

2000 Mathematics Subject Classification: 11A41, 13G99.


Sea R un dominio de factorización única que tiene a lo sumo un número finito de unidades. Nosotros discutimos la infinitud de elementos primos en R cuando R es arbitrario y cuando R satisface la siguiente propiedad: si f y g son polinomios con coeficientes en R tales que f(r) divide g(r) para todo rε R con f(r)≠ 0, entonces g=0 ó grado(f) ≤ grado(g).

Palabras clave: Dominios de factorización única, elementos primos.

Texto completo disponible en PDF


[1] M. Artin, Algebra, Second edn, Prentice Hall, 2011.         [ Links ]

[2] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.         [ Links ]

[3] D. M. Burton, Elementary Number Theory, Fifth edn, McGraw Hill, 2002.         [ Links ]

[4] L. F. Cáceres and J. A. Vélez-Marulanda, `On Certain Divisibility Property of Polynomials over Integral Domains´, J. Math. Research. 3, 3 (2011), 28-31.         [ Links ]

[5] D. S. Dummit and R. M. Foote, Abstract Algebra, Third edn, John Wiley & Sons Inc., 2004.         [ Links ]

[6] B. Fine and G. Rosenberger, Number Theory: An Introduction via the Distribution of Primes, Birkhäuser, 2007.         [ Links ]

[7] H. Gunji and D. L. McQuillan, `On Rings with Certain Divisibility Property´, Michigan. Math. J. 22, 4 (1976), 289-299.         [ Links ]

[8] T. W. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer, 1974.         [ Links ]

[9] I. Kaplansky, `Elementary Divisors and Modules´, Trans. Amer. Math. Soc. 66, (1949), 464-491.         [ Links ]

[10] I. Kaplansky, Commutative Rings, Polygonal Publishing House, 1994.         [ Links ]

[11] D. Lorenzini, An Invitation to Arithmetic Geometry, Graduate Studies in Mathematics Volume 9, American Mathematical Society, 1996.         [ Links ]

[12] W. Narkiewicz, Polynomial Mappings, Lecture Notes in Mathematics 1600, Springer, 1995.         [ Links ]

(Recibido en mayo de 2013. Aceptado en julio de 2013)

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

    AUTHOR  = {Cáceres-Duque, Luis F. and Vélez-Marulanda, José A.},
    TITLE   = {{On the Infinitude of Prime Elements}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2013},
    volume  = {47},
    number  = {2},
    pages   = {167--179}