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Title
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Rings of integer-valued polynomials and derivatives.
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Creator
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Villanueva, Yuri., Charles E. Schmidt College of Science, Department of Mathematical Sciences
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Abstract/Description
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For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c...
Show moreFor D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field.
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Date Issued
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2012
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PURL
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http://purl.flvc.org/FAU/3356899
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Subject Headings
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Rings of integers, Ideals (Algebra), Polynomials, Arithmetic algebraic geometry, Categories (Mathematics), Commutative algebra
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Format
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Document (PDF)
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Title
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Weakly integrally closed domains and forbidden patterns.
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Creator
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Hopkins, Mary E., Charles E. Schmidt College of Science, Department of Mathematical Sciences
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Abstract/Description
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An integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally...
Show moreAn integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every finite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed.
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Date Issued
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2009
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PURL
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http://purl.flvc.org/FAU/199327
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Subject Headings
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Mathematical analysis, Algebra, Homological, Monoids, Categories (Mathematics), Semigroup algebras
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Format
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Document (PDF)
