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ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

Year 2020, Volume: 27 Issue: 27, 206 - 217, 07.01.2020
https://doi.org/10.24330/ieja.663071

Abstract

We introduce the notion of ultra star operation on ultraproduct of integral domains as a map from the set of induced ideals into the set of induced ideals satisfying the traditional properties of star operations. A case of special interest is the construction of an ultra star operation on the ultraproduct of integral domains $R_i$'s from some given star operations $\star_i$ on $R_i$'s. We provide the ultra $b$-operation and the ultra $v$-operation. Given an arbitrary star operation $\star$ on the ultraproduct of some integral domains, we pose the problem of whether the restriction of $\star$ to the set of induced ideals is necessarily an ultra star operation. We show that the ultraproduct of integral domains $R_i$'s is a $\star$-Pr\"{u}fer domain if and only if $R_i$ is a $\star_i$-Pr\"{u}fer domain for $\mathcal{U}$-many $i$.

References

  • D. D. Anderson, D. F. Anderson, M. Fontana and M. Zafrullah, On $v$-domains and star operations, Comm. Algebra, 37(9) (2009), 3018-3043.
  • D. F. Anderson, M. Fontana and M. Zafrullah, Some remarks on Prufer $\star$-multiplication domains and class groups, J. Algebra, 319(1) (2008), 272-295.
  • G. W. Chang, Prufer $\ast$-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra, 319(1) (2008), 309-319.
  • M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings, and related semistar operations, Comm. Algebra, 31(10) (2003), 4775-4805.
  • R. Gilmer, Multiplicative Ideal Theory, Corrected reprint of the 1972 edition, Queen's Papers in Pure and Applied Mathematics, 90, Queen's University, Kingston, ON, 1992.
  • E. G. Houston, S. B. Malik and J. L. Mott, Characterizations of $\star$-multiplication domains, Canad. Math. Bull., 27(1) (1984), 48-52.
  • C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006.
  • S. Malik, J. L. Mott and M. Zafrullah, On t-invertibility, Comm. Algebra, 16(1) (1988), 149-170.
  • J. L. Mott, B. Nashier and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra, 18(5) (1990), 1569-1583.
  • B. Olberding and S. Saydam, Ultraproducts of commutative rings, Commutative Ring Theory and Applications (Fez, 2001), Lecture Notes in Pure and Appl. Math., Dekker, New York, 231 (2003), 369-386.
  • B. Olberding and J. Shapiro, Prime ideals in ultraproducts of commutative rings, J. Algebra, 285(2) (2005), 768-794.
  • H. Prufer, Untersuchen uber Teilbarkeitseigenshafen in Korpern, J. Reine Angew. Math., 168 (1932), 1-36.
  • H. Schoutens, The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics, 1999, Springer-Verlag, Berlin, 2010.
Year 2020, Volume: 27 Issue: 27, 206 - 217, 07.01.2020
https://doi.org/10.24330/ieja.663071

Abstract

References

  • D. D. Anderson, D. F. Anderson, M. Fontana and M. Zafrullah, On $v$-domains and star operations, Comm. Algebra, 37(9) (2009), 3018-3043.
  • D. F. Anderson, M. Fontana and M. Zafrullah, Some remarks on Prufer $\star$-multiplication domains and class groups, J. Algebra, 319(1) (2008), 272-295.
  • G. W. Chang, Prufer $\ast$-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra, 319(1) (2008), 309-319.
  • M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings, and related semistar operations, Comm. Algebra, 31(10) (2003), 4775-4805.
  • R. Gilmer, Multiplicative Ideal Theory, Corrected reprint of the 1972 edition, Queen's Papers in Pure and Applied Mathematics, 90, Queen's University, Kingston, ON, 1992.
  • E. G. Houston, S. B. Malik and J. L. Mott, Characterizations of $\star$-multiplication domains, Canad. Math. Bull., 27(1) (1984), 48-52.
  • C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006.
  • S. Malik, J. L. Mott and M. Zafrullah, On t-invertibility, Comm. Algebra, 16(1) (1988), 149-170.
  • J. L. Mott, B. Nashier and M. Zafrullah, Contents of polynomials and invertibility, Comm. Algebra, 18(5) (1990), 1569-1583.
  • B. Olberding and S. Saydam, Ultraproducts of commutative rings, Commutative Ring Theory and Applications (Fez, 2001), Lecture Notes in Pure and Appl. Math., Dekker, New York, 231 (2003), 369-386.
  • B. Olberding and J. Shapiro, Prime ideals in ultraproducts of commutative rings, J. Algebra, 285(2) (2005), 768-794.
  • H. Prufer, Untersuchen uber Teilbarkeitseigenshafen in Korpern, J. Reine Angew. Math., 168 (1932), 1-36.
  • H. Schoutens, The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics, 1999, Springer-Verlag, Berlin, 2010.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Olivier A. Heubo-kwegna This is me

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

Cite

APA Heubo-kwegna, O. A. (2020). ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS. International Electronic Journal of Algebra, 27(27), 206-217. https://doi.org/10.24330/ieja.663071
AMA Heubo-kwegna OA. ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS. IEJA. January 2020;27(27):206-217. doi:10.24330/ieja.663071
Chicago Heubo-kwegna, Olivier A. “ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS”. International Electronic Journal of Algebra 27, no. 27 (January 2020): 206-17. https://doi.org/10.24330/ieja.663071.
EndNote Heubo-kwegna OA (January 1, 2020) ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS. International Electronic Journal of Algebra 27 27 206–217.
IEEE O. A. Heubo-kwegna, “ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS”, IEJA, vol. 27, no. 27, pp. 206–217, 2020, doi: 10.24330/ieja.663071.
ISNAD Heubo-kwegna, Olivier A. “ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS”. International Electronic Journal of Algebra 27/27 (January 2020), 206-217. https://doi.org/10.24330/ieja.663071.
JAMA Heubo-kwegna OA. ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS. IEJA. 2020;27:206–217.
MLA Heubo-kwegna, Olivier A. “ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS”. International Electronic Journal of Algebra, vol. 27, no. 27, 2020, pp. 206-17, doi:10.24330/ieja.663071.
Vancouver Heubo-kwegna OA. ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS. IEJA. 2020;27(27):206-17.