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2013 SUBMAXIMAL INTEGRAL DOMAINS
A. Azarang
Taiwanese J. Math. 17(4): 1395-1412 (2013). DOI: 10.11650/tjm.17.2013.2332

Abstract

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of $R$, or $R$ has zero characteristic and there exists a natural number $n\gt 1$ such that $\frac{1}{n}\in R$, then $R$ has a maximal subring. It is shown that if $R$ is a reduced ring with $|R|\gt 2^{2^{\aleph_0}}$ or $J(R)\neq 0$, then any $R$-algebra has a maximal subring. Residually finite rings without maximal subrings are fully characterized. It is observed that every uncountable $UFD$ has a maximal subring. The existence of maximal subrings in a noetherian integral domain $R$, in relation to either the cardinality of the set of divisors of some of its elements or the height of its maximal ideals, is also investigated.

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A. Azarang. "SUBMAXIMAL INTEGRAL DOMAINS." Taiwanese J. Math. 17 (4) 1395 - 1412, 2013. https://doi.org/10.11650/tjm.17.2013.2332

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1300.13007
MathSciNet: MR3085517
Digital Object Identifier: 10.11650/tjm.17.2013.2332

Subjects:
Primary: 13B02, 13E05, 13G05

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 4 • 2013
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