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FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?

Year 2020, Volume: 27 Issue: 27, 43 - 60, 07.01.2020

Ryan Gipson Hamid Kulosman

https://doi.org/10.24330/ieja.662949
Cited By: 2

Abstract

We characterize the Puiseux monoids $M$ for which the irreducible and the prime elements in the monoid ring $F[X;M]$, where $F$ is a field, coincide. We present a diagram of implications between some types of Puiseux monoids, with a precise position of the monoids $M$ with this property.

Keywords

Puiseux monoid monoid domain AP domain atomic domain PC domain gcd/lcm property height (0; 0; 0; : : : ) Prüfer monoid integrally closed monoid

References

  • D. F. Anderson, Robert Gilmer's work on semigroup rings, in \Multiplicative Ideal Theory In Commutative Algebra, A tribute to the work of Robert Gilmer" (J. Brewer et al. (Eds)), Springer Science+Business media, LLC, (2006), 21-37.
  • D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra, 69(1) (1990), 1-19.
  • K. E. Aubert, Theory of x-ideals, Acta Math., 107 (1962), 1-52.
  • S. T. Chapman, F. Gotti and M. Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, to appear in Comm. Algebra, DOI: 10.1080/00927872.2019.1646269; also arXiv1904.00219.
  • K. Christensen, R. Gipson and H. Kulosman, Irreducibility of certain binomials in semigroup rings for nonnegative rational monoids, Int. Electron. J. Algebra, 24 (2018), 50-61.
  • K. Christensen, R. Gipson and H. Kulosman, A new characterization of principal ideal domains, arXiv 1805.10374v1 [math.AC].
  • P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc., 64 (1968), 251-264.
  • P. M. Cohn, Algebra, Vol. I, Second Edition, John Wiley & Sons Sons, Ltd., Chichester, 1982.
  • J. Coykendall and F. Gotti, On the atomicity of monoid algebras, J. Algebra, 539 (2019), 138-151.
  • J. Coykendall and B. Johnson Mammenga, An embedding theorem, J. Algebra, 325 (2011), 177-185.
  • J. Coykendall and M. Zafrullah, AP-domains and unique factorization, J. Pure Appl. Algebra, 189 (2004), 27-35.
  • R. C. Daileda, A non-UFD integral domains in which irreducibles are prime, preprint, http://ramanujan.math.trinity.edu/daileda/teach/m4363s07/nonufd. pdf.
  • L. Fuchs, In nite Abelian Groups, Vol. II, Pure and Applied Mathematics, Vol. 36-II, Academic Press, New York-London, 1973.
  • A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, 278, Chapman & Hall/CRC, Boca Raton, FL, 2006.
  • A. Geroldinger and W. Hassler, Local tameness of v-noetherian monoids, J. Pure Appl. Algebra, 212(6) (2008), 1509-1524.
  • A. Geroldinger and W. Hassler, Arithmetic of Mori domains and monoids, J. Algebra, 319(8) (2008), 3419-3463.
  • A. Geroldinger and A. Reinhart, The monotone catenary degree of monoids of ideals, Internat. J. Algebra Comput., 29(3) (2019), 419-457.
  • R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972.
  • R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
  • R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J., 21 (1974), 6586.
  • R. Gilmer and T. Parker, Semigroup rings as Prufer rings, Duke Math. J., 41 (1974), 219-230.
  • R. Gipson and H. Kulosman, Atomic and AP semigroup rings F[X;M], where M is a submonoid of the additive monoid of nonnegative rational numbers, Int. Electron. J. Algebra, 22 (2017), 133-146.
  • F. Gotti, On the atomic structure of Puiseux monoids, J. Algebra Appl., 16(7) (2017), 1750126 (20 pp).
  • F. Gotti, Puiseux monoids and transfer homomorphisms, J. Algebra, 516 (2018), 95-114.
  • M. Gotti, On the local k-elasticities of Puiseux monoids, Internat. J. Algebra Comput., 29(1) (2019), 147-158.
  • F. Gotti and C. O'Neill, The elasticity of Puiseux monoids, to appear in J. Commut. Algebra, https://projecteuclid.org/euclid.jca/1523433696; also arXiv 1703.04207v1 [math.AC].
  • A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc., 75 (1974), 321-329.
  • P. A. Grillet, Commutative Semigroups, Advances in Mathematics (Dordrecht), 2, Kluwer Academic Publishers, Dordrecht, 2001.
  • F. Halter-Koch, Ideal Systems, An Introduction to Multiplicative Ideal Theory, Monographs and Textbooks in Pure and Applied Mathematics, 211, Marcel Dekker, Inc., New York, 1998.
  • H. C. Hutchins, Examples of Commutative Rings, Polygonal Publ. House, Washington, N. J., 1981.
  • P. Jaffard, Les Systems des Ideaux, Dunod, 1960.
  • I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago and London, 1974.
  • H. Kulosman, A new simple example of an atomic domain which is not ACCP, Adv. Algebra, 12 (2019), 1-7.
  • R. Matsuda, Torsion-free abelian group rings III, Bull. Fac. Sci., Ibaraki Univ. Ser. A, 9 (1977), 1-49.
  • D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, Cambridge, 1968.
  • M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra, 15(9) (1987), 1895-1920.

Year 2020, Volume: 27 Issue: 27, 43 - 60, 07.01.2020

Ryan Gipson Hamid Kulosman

https://doi.org/10.24330/ieja.662949
Cited By: 2

Abstract

References

  • D. F. Anderson, Robert Gilmer's work on semigroup rings, in \Multiplicative Ideal Theory In Commutative Algebra, A tribute to the work of Robert Gilmer" (J. Brewer et al. (Eds)), Springer Science+Business media, LLC, (2006), 21-37.
  • D. D. Anderson, D. F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra, 69(1) (1990), 1-19.
  • K. E. Aubert, Theory of x-ideals, Acta Math., 107 (1962), 1-52.
  • S. T. Chapman, F. Gotti and M. Gotti, Factorization invariants of Puiseux monoids generated by geometric sequences, to appear in Comm. Algebra, DOI: 10.1080/00927872.2019.1646269; also arXiv1904.00219.
  • K. Christensen, R. Gipson and H. Kulosman, Irreducibility of certain binomials in semigroup rings for nonnegative rational monoids, Int. Electron. J. Algebra, 24 (2018), 50-61.
  • K. Christensen, R. Gipson and H. Kulosman, A new characterization of principal ideal domains, arXiv 1805.10374v1 [math.AC].
  • P. M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc., 64 (1968), 251-264.
  • P. M. Cohn, Algebra, Vol. I, Second Edition, John Wiley & Sons Sons, Ltd., Chichester, 1982.
  • J. Coykendall and F. Gotti, On the atomicity of monoid algebras, J. Algebra, 539 (2019), 138-151.
  • J. Coykendall and B. Johnson Mammenga, An embedding theorem, J. Algebra, 325 (2011), 177-185.
  • J. Coykendall and M. Zafrullah, AP-domains and unique factorization, J. Pure Appl. Algebra, 189 (2004), 27-35.
  • R. C. Daileda, A non-UFD integral domains in which irreducibles are prime, preprint, http://ramanujan.math.trinity.edu/daileda/teach/m4363s07/nonufd. pdf.
  • L. Fuchs, In nite Abelian Groups, Vol. II, Pure and Applied Mathematics, Vol. 36-II, Academic Press, New York-London, 1973.
  • A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations, Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, 278, Chapman & Hall/CRC, Boca Raton, FL, 2006.
  • A. Geroldinger and W. Hassler, Local tameness of v-noetherian monoids, J. Pure Appl. Algebra, 212(6) (2008), 1509-1524.
  • A. Geroldinger and W. Hassler, Arithmetic of Mori domains and monoids, J. Algebra, 319(8) (2008), 3419-3463.
  • A. Geroldinger and A. Reinhart, The monotone catenary degree of monoids of ideals, Internat. J. Algebra Comput., 29(3) (2019), 419-457.
  • R. Gilmer, Multiplicative Ideal Theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972.
  • R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984.
  • R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J., 21 (1974), 6586.
  • R. Gilmer and T. Parker, Semigroup rings as Prufer rings, Duke Math. J., 41 (1974), 219-230.
  • R. Gipson and H. Kulosman, Atomic and AP semigroup rings F[X;M], where M is a submonoid of the additive monoid of nonnegative rational numbers, Int. Electron. J. Algebra, 22 (2017), 133-146.
  • F. Gotti, On the atomic structure of Puiseux monoids, J. Algebra Appl., 16(7) (2017), 1750126 (20 pp).
  • F. Gotti, Puiseux monoids and transfer homomorphisms, J. Algebra, 516 (2018), 95-114.
  • M. Gotti, On the local k-elasticities of Puiseux monoids, Internat. J. Algebra Comput., 29(1) (2019), 147-158.
  • F. Gotti and C. O'Neill, The elasticity of Puiseux monoids, to appear in J. Commut. Algebra, https://projecteuclid.org/euclid.jca/1523433696; also arXiv 1703.04207v1 [math.AC].
  • A. Grams, Atomic rings and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc., 75 (1974), 321-329.
  • P. A. Grillet, Commutative Semigroups, Advances in Mathematics (Dordrecht), 2, Kluwer Academic Publishers, Dordrecht, 2001.
  • F. Halter-Koch, Ideal Systems, An Introduction to Multiplicative Ideal Theory, Monographs and Textbooks in Pure and Applied Mathematics, 211, Marcel Dekker, Inc., New York, 1998.
  • H. C. Hutchins, Examples of Commutative Rings, Polygonal Publ. House, Washington, N. J., 1981.
  • P. Jaffard, Les Systems des Ideaux, Dunod, 1960.
  • I. Kaplansky, Commutative Rings, Revised Edition, The University of Chicago Press, Chicago and London, 1974.
  • H. Kulosman, A new simple example of an atomic domain which is not ACCP, Adv. Algebra, 12 (2019), 1-7.
  • R. Matsuda, Torsion-free abelian group rings III, Bull. Fac. Sci., Ibaraki Univ. Ser. A, 9 (1977), 1-49.
  • D. G. Northcott, Lessons on Rings, Modules and Multiplicities, Cambridge University Press, Cambridge, 1968.
  • M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra, 15(9) (1987), 1895-1920.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ryan Gipson This is me

Hamid Kulosman This is me

Publication Date January 7, 2020
Published in Issue Year 2020 Volume: 27 Issue: 27

Cite

APA Gipson, R., & Kulosman, H. (2020). FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?. International Electronic Journal of Algebra, 27(27), 43-60. https://doi.org/10.24330/ieja.662949
AMA Gipson R, Kulosman H. FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?. IEJA. January 2020;27(27):43-60. doi:10.24330/ieja.662949
Chicago Gipson, Ryan, and Hamid Kulosman. “FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 43-60. https://doi.org/10.24330/ieja.662949.
EndNote Gipson R, Kulosman H (January 1, 2020) FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?. International Electronic Journal of Algebra 27 27 43–60.
IEEE R. Gipson and H. Kulosman, “FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?”, IEJA, vol. 27, no. 27, pp. 43–60, 2020, doi: 10.24330/ieja.662949.
ISNAD Gipson, Ryan - Kulosman, Hamid. “FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?”. International Electronic Journal of Algebra 27/27 (January 2020), 43-60. https://doi.org/10.24330/ieja.662949.
JAMA Gipson R, Kulosman H. FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?. IEJA. 2020;27:43–60.
MLA Gipson, Ryan and Hamid Kulosman. “FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 43-60, doi:10.24330/ieja.662949.
Vancouver Gipson R, Kulosman H. FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?. IEJA. 2020;27(27):43-60.

Cited By

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https://doi.org/10.1080/00927872.2021.1949018