1999 Fiscal Year Final Research Report Summary
Considerations on seminormal semigroups and certain seminormal rings
| Project/Area Number |
09640027
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Aichi University of Education |
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Principal Investigator |
KANEMITSU Mitsuo Aichi University of Education, Faculty of Education, Professor, 教育学部, 教授 (60024014)
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| Co-Investigator(Kenkyū-buntansha) |
YASUMOTO Taichi Aichi University of Education, Faculty of Education, Assitant Professor, 教育学部, 助教授 (00231647)
ODANI Kenzi Aichi University of Education, Faculty of Education, Professor, 教育学部, 助教授 (60273299)
FURUKAWA Yasukimi Aichi University of Education, Faculty of Education, Professor, 教育学部, 教授 (90024033)
MATSUDA Ryuki Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (10006934)
YOSHIDA Ken-ichi Okayama University of Science, Faculty of Science, Professor, 理学部, 教授 (60028264)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Seminormal semigroup / integral oversemigroup / prime divisor / flat ring-extension / LCM-stable extension / valuation semigroup / valuation ideal / anti-integral extension |
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| Research Abstract |
All rings are assumed to be communicative with identify. We assume that all semigroups, written adaptively, are non-zero, commutative, torsion-free cancellative semigroups with O. We obtained the following results.
(1) We considered the properties of seminormal semigroups.
For a semigroups S with dim(S) = 1, we have the following theorems.
1. If S is seminormal, then each integral oversemigroup of S is seminormal.
2. Each oversemigroup of S is seminormal is and only if both S is seminormal and the integral closure S' of S is a valuation semigroup.
(2) Let R be a Noetherian domain with the quotient field K. Let α be an algebraic element over K such that α satisfies certain conditions (that is, α is an anti-integral criterion that R[α] is a flat extension over R by using some ideal whether or no it is invertible. Also, if R[α] is LCM-stable over R under certain conditions then R[α] is flat over R.
(3) We treat a problem which determine the prime divisors concerning the contraction of typical ideals in simple ring-extensions. We perfectly determined determined this problem under certain conditions.
(4) We investigate some prime ideals of a polynomial semigroup S[X]. For example, we prove that any radical ideals of a Noetherian monoid S is the intersection of a finite number of prime ideals.
(5) We give some properties of cancellation of ideals in a semigroup and also we give some characterizations of valuation semigroups.
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