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