Rocky Mountain Journal of Mathematics

Content formulas for power series and Krull domains

Huayu Yin, Youhua Chen, and Xiaosheng Zhu

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Abstract

Let $R$ be an integral domain with quotient field $K$, and let $X$ be an indeterminate over $R$. In this paper, we consider content formulae for power series in terms of $*$-operations for PVMDs, Krull domains and Dedekind domains, where $*$ is the star-operation, $d$, $w$, $t$, or $v$. We prove that $R$ is a Krull domain if and only if $c(f/g)_w=(c(f)c(g)^{-1})_w$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal if and only if $c(f/g)_t=(c(f)c(g)^{-1})_t$ for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal, and $R$ is a Dedekind domain if and only if for all $f,g\in R[[X]]^*$ with $c(f/g)$ a fractional ideal, $c(f/g)=c(f)c(g)^{-1}$.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 6 (2016), 2077-2088.

Dates
First available in Project Euclid: 4 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1483520439

Digital Object Identifier
doi:10.1216/RMJ-2016-46-6-2077

Mathematical Reviews number (MathSciNet)
MR3591273

Zentralblatt MATH identifier
1362.13005

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory
Secondary: 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F25: Formal power series rings [See also 13J05]

Keywords
Content completely integrally closed Krull domain Dedekind domain

Citation

Yin, Huayu; Chen, Youhua; Zhu, Xiaosheng. Content formulas for power series and Krull domains. Rocky Mountain J. Math. 46 (2016), no. 6, 2077--2088. doi:10.1216/RMJ-2016-46-6-2077. https://projecteuclid.org/euclid.rmjm/1483520439


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References

  • D.D. Anderson, GCD domains, Gauss' lemma, and contents of polynomials, in Non-Noetherian commutative ring theory, mathematics and its applications, S.T. Chapman and S. Glaz, eds., Volume 520, Kluwer Academic Publishers, Dordrecht, 2000.
  • D.D. Anderson and B.G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), 82–94.
  • ––––, Formally integrally closed domains and the rings $R((X))$ and $R\{\{X\}\}$, J. Algebra 200 (1998), 347–362.
  • D.D. Anderson, D.J. Kwak and M. Zafrullah, Agreeable domains, Comm. Alg. 23 (1995), 4861–4883.
  • D.F. Anderson, M. Fontana and M. Zafrullah, Some remarks on Prüfer $\star$-multiplication domains and class groups, J. Algebra 319 (2008), 272–295.
  • R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
  • B.G. Kang, Prüfer $v$-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), 151–170.
  • M.H. Park, Group rings and semigroup rings over strong Mori domains, J. Pure Appl. Algebra 163 (2001), 301–318.
  • J. Querre, Idéaux divisoriels d'un anneau de polynômes, J. Algebra 64 (1980), 270–284.
  • F.G. Wang and R.L. McCasland, On $w$-modules over strong Mori domains, Comm. Alg. 25 (1997), 1285–1306.
  • ––––, On strong Mori domains, J. Pure Appl. Algebra 135 (1999), 155–165.