Maximal subrings of division rings

TL;DR

This paper explores the structure, existence, and properties of maximal subrings in division rings, revealing conditions under which they exist and their algebraic and valuation characteristics.

Contribution

It provides new criteria for the existence and structure of maximal subrings in division rings, including their relation to algebraic elements and centralizers.

Findings

If a division ring has a noncentral algebraic element, it has a maximal subring.

A non-commutative division ring either has a maximal subring or has large dimension over its center.

Maximal subrings can be characterized as centralizers of algebraic elements or certain valuation rings.

Abstract

The structure and the existence of maximal subrings in division rings are investigated. We see that if $R$ is a maximal subring of a division ring $D$ with center $F$ and $N(R)\neq U(R)\cup \{0\}$, where $N(R)$ is the normalizer of $R$ in $D$, then either $R$ is a division ring with $[D:R]_l=[D:R]_r$ is finite or $R$ is an Ore $G$-domain with certain properties. In particular, if $F\subsetneq C_D(R)$, the centralizer of $R$ in $D$, then $R=C_D(\beta)$ is a division ring, for each $\beta\in C_R(R)\setminus F$, $[D:R]_l$ is finite if and only if $\beta$ is algebraic over $F$, $[D:R]_l=[D:R]_r=[F[\beta]:F]$ and $C_R(R)=F[\beta]$. On the other hand if $R$ does not contains $F$, then $R\cap F=C_R(R)$ is a maximal subring of $F$. Consequently, if a division ring $D$ has a noncentral element which is algebraic over the center of $D$, then $D$ has a maximal subring. In particular, we prove that if $D$ is a non-commutative division ring with center $F$, then either $D$ has a maximal subring or $dim_F(D)\geq |F|$. We study when a maximal subring of a division ring is a left duo ring or certain valuation rings. Finally, we prove that if $D$ is an existentially complete division ring over a field $K$, then $D$ has a maximal subring of the form $C_D(x)$ where $D$ is finite over it. Moreover, if $R$ is a maximal subring of $D$ with $K\subsetneq C_R(R)$, then $R=C_D(x)$ for some $x\in D\setminus K$, which is algebraic over $K$.