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A primrose path from Krull to Zorn

Marcel Erné (1995)

Commentationes Mathematicae Universitatis Carolinae

Given a set X of “indeterminates” and a field F , an ideal in the polynomial ring R = F [ X ] is called conservative if it contains with any polynomial all of its monomials. The map S ↦ R S yields an isomorphism between the power set P ( X ) and the complete lattice of all conservative prime ideals of R . Moreover, the members of any system S ⊆ P ( X ) of finite character are in one-to-one correspondence with the conservative prime ideals contained in P S = ⋃ { R S : S ∈ S } , and the maximal members of S correspond to the maximal ideals contained in...

Anneaux de Goldman

Jean Guérindon (1969/1970)

Séminaire Dubreil. Algèbre et théorie des nombres

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R , then R contains one of them under various conditions.

Equations for the set of overrings of normal rings and related ring extensions

Mabrouk Ben Nasr, Ali Jaballah (2023)

Czechoslovak Mathematical Journal

We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.

Finite dimensional rings of quotients.

H. Leroy Hutson (1993)

Publicacions Matemàtiques

In this paper we characterize commutative rings with finite dimensional classical ring of quotients. To illustrate the diversity of behavior of these rings we examine the case of local rings and FPF rings. Our results extend earlier work on rings with zero-dimensional rings of quotients.

Maximal non valuation domains in an integral domain

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

Let R be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring R of an integral domain S is called a maximal non valuation domain in S if R is not a valuation subring of S , and for any ring T such that R ⊂ T ⊂ S , T is a valuation subring of S . For a local domain S , the equivalence of an integrally closed maximal non VD in S and a maximal non local subring of S is established. The relation between dim ( R , S ) and the number...

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