Frontiers of Mathematics in China >

2016 , Vol. 11 >Issue 4: 829 - 844

DOI: https://doi.org/10.1007/s11464-016-0554-7

RESEARCH ARTICLE

Valuation ideals and primary w-ideals

  • Gyu Whan CHANG 1,2 ,
  • Hwankoo KIM , 1,2
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  • 1. Department of Mathematics Education, Incheon National University, Incheon 406-772, Republic of Korea
  • 2. Department of Information Security, Hoseo University, Asan 336-795, Republic of Korea

Received date: 09 Nov 2015

Accepted date: 29 Jan 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Let D be an integral domain, V(D) (resp., t-V(D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and Nv= {f ∈ D[X] | c(f)v=D}. In this paper, we study integral domains D in which w-P(D) ⊆ t-V(D), t-V(D) ⊆ w-P(D), or t-V(D) = w-P(D). We also study the relationship between t-V(D) and V(D[X]Nv), and characterize when t-V(A + XB[X]) ⊆w-P(A + XB[X]) holds for a proper extension A ⊂ B of integral domains.

Key words： t-Valuation ideal; primary w-ideal; PvMD; UMT-domain; D[X]Nv

Cite this article

Gyu Whan CHANG , Hwankoo KIM . Valuation ideals and primary w-ideals[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 829 -844 . DOI: 10.1007/s11464-016-0554-7

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