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Graph theoretic characterizations of maximal non-valuation subrings of a field

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Abstract

We establish several new characterizations of maximal non-valuation subrings of a field involving several concepts of commutative algebra related to the set of prime ideals and the set of overrings. For example we show that an integral domain R of finite dimension d is a maximal non-valuation subring of a field if, and only if R is either integrally closed with a set of overrings isomorphic to a kite-graph of dimension d + 1, or is non-integrally closed with a chained set of overrings of dimension d + 1.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Univesity of Sharjah, Sharjah, UAE

    Ali Jaballah

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  1. Ali Jaballah
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Correspondence to Ali Jaballah.

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Jaballah, A. Graph theoretic characterizations of maximal non-valuation subrings of a field. Beitr Algebra Geom 54, 111–120 (2013). https://doi.org/10.1007/s13366-012-0101-y

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  • DOI: https://doi.org/10.1007/s13366-012-0101-y

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