Submaximal Integral Domains

TL;DR

This paper investigates conditions under which various classes of rings, including integral domains and algebras, possess maximal subrings, providing new existence results and characterizations based on algebraic and cardinality properties.

Contribution

It establishes new criteria for the existence of maximal subrings in different types of rings, including UFDs, reduced rings, and noetherian domains, with novel characterizations and conditions.

Findings

Uncountable UFDs always have maximal subrings.

Rings with non-algebraic units over their prime subring have maximal subrings.

Reduced rings with large cardinality or non-zero Jacobson radical have maximal subrings.

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.