1University of Puerto Rico at Mayagüez, Mayagüez, PR, USA. Email: luis.caceres1@upr.edu
2Valdosta State University, Valdosta, GA, USA. Email: javelezmarulanda@valdosta.edu
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.
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.
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References
]]>[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 ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCMv47n2a04,
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}
}