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arXiv · 1208.5298

Submaximal Integral Domains

A. Azarang
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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 $n>1$ such that $\frac{1}{n}\in R$, then $R$ has a maximal subring. It is shown that if $R$ is a reduced ring with $|R|>2^{2^{\aleph_0}}$ 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. 2012-08-27. Submaximal Integral Domains. https://arxiv.org/abs/1208.5298

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