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### Weakly integrally closed domains and forbidden patterns

Date Issued:
2009
Summary:
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 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.
 Title: Name(s): Weakly integrally closed domains and forbidden patterns. 74 views 12 downloads Hopkins, Mary E. Charles E. Schmidt College of Science Department of Mathematical Sciences text Electronic Thesis Or Dissertation 2009 Florida Atlantic University electronic v, 39 p. English 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 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. 351399009 (oclc), 199327 (digitool), FADT199327 (IID), fau:3003 (fedora) by Mary E. Hopkins.Thesis (Ph.D.)--Florida Atlantic University, 2009.Includes bibliography.Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web. Mathematical analysisAlgebra, HomologicalMonoidsCategories (Mathematics)Semigroup algebras http://purl.flvc.org/FAU/199327 http://rightsstatements.org/vocab/InC/1.0/ FAU