DC Field | Value | Language |
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dc.contributor.author | Park, Mi Hee | - |
dc.contributor.author | Kang, Byung Gyun. | - |
dc.contributor.author | Phan Thanh Toan | - |
dc.date.accessioned | 2019-04-07T16:56:00Z | - |
dc.date.available | 2019-04-07T16:56:00Z | - |
dc.date.created | 2018-05-15 | - |
dc.date.issued | 2018-08 | - |
dc.identifier.issn | 0022-4049 | - |
dc.identifier.uri | https://oasis.postech.ac.kr/handle/2014.oak/95718 | - |
dc.description.abstract | Let R[X] be the power series ring over a commutative ring R with identity. For f is an element of R[X], let A(f) denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Priifer domain and if g is an element of R[X] such that A(g) is locally finitely generated (or equivalently locally principal), then a Dedekind-Mertens type formula holds for g, namely A(f)(2)A(g) = A(f)A(fg) for all f is an element of R[X]. More generally for a Prefer domain R, we prove the content formula (A(f)A(g))(2) = (A(f)A(g))A(fg) for all f, g is an element of R[X]. As a consequence it is shown that an integral domain R is completely integrally closed if and only if (A(f)A(g))(v) = (A(fg))(v) for all nonzero f,g is an element of R[X], which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if (A(f)A(g))(v) = (A(fg))(v) for all nonzero f,g is an element of R[X], where R[X] is the polynomial ring over R. For a ring R and g is an element of R[X], if A(g) is not locally finitely generated, then there may be no positive integer k such that A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. Assuming that the locally minimal number of generators of A(g) is k +1, Epstein and Shapiro posed a question about the validation of the formula A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of A(g) is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of A(g) is k + 1, then A(f)(k+1)A(g) = A(f)(k)A(fg) for all f is an element of R[X]. As a consequence we show that if A(g) is finitely generated (in particular if g is an element of R[X]), then there exists a nonnegative integer k such that A(f)(k+1)A(g)= A(f)(k)A(fg) for all f is an element of R[X]. (C) 2017 Elsevier B.V. All rights reserved. | - |
dc.language | English | - |
dc.publisher | ELSEVIER SCIENCE BV | - |
dc.relation.isPartOf | JOURNAL OF PURE AND APPLIED ALGEBRA | - |
dc.title | Dedekind-Mertens lemma and content formulas in power series rings | - |
dc.type | Article | - |
dc.identifier.doi | 10.1016/j.jpaa.2017.09.013 | - |
dc.type.rims | ART | - |
dc.identifier.bibliographicCitation | JOURNAL OF PURE AND APPLIED ALGEBRA, v.222, no.8, pp.2299 - 2309 | - |
dc.identifier.wosid | 000428827400019 | - |
dc.citation.endPage | 2309 | - |
dc.citation.number | 8 | - |
dc.citation.startPage | 2299 | - |
dc.citation.title | JOURNAL OF PURE AND APPLIED ALGEBRA | - |
dc.citation.volume | 222 | - |
dc.contributor.affiliatedAuthor | Kang, Byung Gyun. | - |
dc.identifier.scopusid | 2-s2.0-85029717668 | - |
dc.description.journalClass | 1 | - |
dc.description.journalClass | 1 | - |
dc.description.isOpenAccess | N | - |
dc.type.docType | Article | - |
dc.subject.keywordPlus | POLYNOMIALS | - |
dc.relation.journalWebOfScienceCategory | Mathematics, Applied | - |
dc.relation.journalWebOfScienceCategory | Mathematics | - |
dc.description.journalRegisteredClass | scie | - |
dc.description.journalRegisteredClass | scopus | - |
dc.relation.journalResearchArea | Mathematics | - |
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