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Given a set of “indeterminates” and a field , an ideal in the polynomial ring is called conservative if it contains with any polynomial all of its monomials. The map yields an isomorphism between the power set and the complete lattice of all conservative prime ideals of . Moreover, the members of any system of finite character are in one-to-one correspondence with the conservative prime ideals contained in , and the maximal members of correspond to the maximal ideals contained in...
Let be a commutative ring with identity. If a ring is contained in an arbitrary union of rings, then is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in , then contains one of them under various conditions.
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.
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.
Let be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring of an integral domain is called a maximal non valuation domain in if is not a valuation subring of , and for any ring such that , is a valuation subring of . For a local domain , the equivalence of an integrally closed maximal non VD in and a maximal non local subring of is established. The relation between and the number...
Let be an integral domain with the quotient field , an indeterminate over and an element of . The Bhargava ring over at is defined to be . In fact, is a subring of the ring of integer-valued polynomials over . In this paper, we aim to investigate the behavior of under localization. In particular, we prove that behaves well under localization at prime ideals of , when is a locally finite intersection of localizations. We also attempt a classification of integral domains ...
For and open in , let be the ring of real valued functions on with the first derivatives continuous. It is shown that for there is with and with . The function and its derivatives are not assumed to be bounded on . The function is constructed using splines based on the Mollifier function. Some consequences about the ring are deduced from this, in particular that .
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