Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterizing when $ R[X]$ is integrally closed


Author: Thomas G. Lucas
Journal: Proc. Amer. Math. Soc. 105 (1989), 861-867
MSC: Primary 13B20; Secondary 13B30
MathSciNet review: 942636
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Unlike the situation when dealing with integral domains, it is not always the case that the polynomial ring $ R\left[ X \right]$ is integrally closed when $ R$ is an integrally closed commutative ring with nonzero zero divisors. In the main theorem it is shown that for an integrally closed reduced ring $ R$, $ R\left[ X \right]$ is not integrally closed if and only if there exists a finitely generated dense ideal $ J$ and an $ R$-module homomorphism $ s \in {\text{Ho}}{{\text{m}}_R}\left( {J,R} \right)$ such that $ s$ is integral over $ R$ and $ s$ is not defined by multiplication by a fixed element of $ R$. As a corollary it is shown that $ R\left[ X \right]$ is integrally closed if and only if $ R$ is integrally closed in $ T\left( {R\left[ X \right]} \right)$, the total quotient ring of $ R\left[ X \right]$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13B20, 13B30

Retrieve articles in all journals with MSC: 13B20, 13B30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0942636-0
PII: S 0002-9939(1989)0942636-0
Keywords: Reduced ring, integrally closed, complete ring of quotients, dense ideal
Article copyright: © Copyright 1989 American Mathematical Society