Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Composite Hurwitz Rings Satisfying the Ascending Chain Condition on Principal Ideals

  • Lim, Jung Wook (Department of Mathematics, Kyungpook National University) ;
  • Oh, Dong Yeol (Department of Mathematics Education, Chosun University)
  • Received : 2015.09.24
  • Accepted : 2016.03.15
  • Published : 2016.12.23
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Abstract

Let $D{\subseteq}E$ be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D, E) and H(D, I) (resp., h(D, E) and h(D, I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D, E) satisfies the ascending chain condition on principal ideals if and only if h(D, E) satisfies the ascending chain condition on principal ideals, if and only if ${\bigcap}_{n{\geq}1}a_1{\cdots}a_nE=(0)$ for each infinite sequence $(a_n)_{n{\geq}1}$ consisting of nonzero nonunits of We also prove that H(D, I) satisfies the ascending chain condition on principal ideals if and only if h(D, I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.

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References

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Cited by

  1. Chain conditions on composite Hurwitz series rings vol.15, pp.1, 2017, https://doi.org/10.1515/math-2017-0097