Maximal subrings up to isomorphism of fields

TL;DR

This paper investigates the classification and properties of maximal subrings up to isomorphism in various fields and rings, revealing conditions for finiteness and structure, especially in characteristic zero and algebraically closed fields.

Contribution

It provides new results on the number and structure of maximal subrings up to isomorphism in fields and rings, including criteria for finiteness and the behavior under automorphisms.

Findings

Fields with zero characteristic have infinitely many maximal subrings up to isomorphism.

Integrally closed maximal subrings of K(x) are all isomorphic if K is algebraically closed.

If K is absolutely algebraic, K(x) has finitely many integrally closed maximal subrings up to isomorphism iff K is algebraically closed.

Abstract

In this paper we study maximal subrings up to isomorphism of fields. It is shown that each field with zero characteristic has infinitely many maximal subrings up to isomorphism. If $K$ is an algebraically closed field and $x$ is an indeterminate over $K$, then we prove that integrally closed maximal subrings of $K(x)$ which contains $K$ are all isomorphic. In particular, if $K$ is an absolutely algebraic field, then $K(x)$ has only finitely many integrally closed maximal subrings up to isomorphism if and only if $K$ is algebraically closed. Also, we show that if $K$ is an absolutely algebraic field then $K$ has only finitely many maximal subrings up to isomorphism if and only if $K$ has only finitely many maximal subrings. We prove that if a commutative ring $R$ with zero characteristic has only finitely many maximal subrings up to isomorphism, then $U(R)\cap\mathbb{P}$ is finite, where $\mathbb{P}$ is the set of natural prime numbers. In particular, if $R$ is a commutative ring with zero characteristic and $Char(R/J(R))\neq 0$, then $R$ has infinitely many maximal subrings up to isomorphism. Maximal subrings up to isomorphism of $K[x]$, $K\times K$ and $K[x]/(x^2)$ for a field $K$ are investigated. If $R$ is a non-field maximal subring of a field $K$ and $\mathcal{A}$ is the set of all maximal subrings of $K$ which are isomorphic to $R$, then we prove that $\mathcal{A}=\{\sigma(R)\ |\ \sigma\in Aut(K)|\}$, in particular $|\mathcal{A}|\leq |Aut(K)|$. Moreover if $K$ is infinite and $|Aut(K)|<|K|$, then $R$ has at least $|K|$-many integrally closed maximal subrings up to isomorphism.