Pairs of commutative rings in which all intermediate rings have the same dimension

• Autores: R. Douglas Chatham, David E. Dobbs
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 33, Nº 3, 2007, págs. 635-647
• Idioma: inglés
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• Resumen

  • If n is a nonnegative integer or infinity, and R is a (commutative unital) ring contained in a (commutative unital) ring T, then (R,T) is said to be an n-dimensional pair if every ring S that both contains R and is contained in T has Krull dimension n. For n greater than zero, examples are given of n-dimensional pairs that are not integral extensions, including an infinite family of examples that are neither LO-pairs nor INC-pairs. The n-dimensional pair property transfers well in constructions involving pullbacks or passage to the associated reduced rings, but this property is not stable under passage to factor domains. Special attention is paid to the n-dimensional pairs whose first coordinate is a Jaffard domain or a residually Jaffard ring. Also, examples are given of infinity-dimensional pairs whose intermediate rings have prime ideal chains of arbitrarily large cardinality; and of a family of n-dimensional pairs arising from minimal overrings.