Maximal subrings of certain non-commutative rings

TL;DR

This paper investigates conditions under which certain non-commutative rings possess maximal subrings, providing various criteria and results for rings integral over their centers, including specific cases like Artinian, Noetherian, and product rings.

Contribution

It establishes new criteria for the existence of maximal subrings in non-commutative rings that are integral over their centers, extending previous results to broader classes of rings.

Findings

Rings integral over their centers either have a maximal subring or specific structural properties.

Left Artinian rings integral over their centers are either countable with a maximal subring or have special properties.

Product rings of rings integral over their centers always have a maximal subring.

Abstract

The existence of maximal subrings in certain non-commutative rings, especially in rings which are integral over their centers, are investigated. We prove that if a ring $T$ is integral over its center, then either $T$ has a maximal subring or $T/J(T)$ is a commutative Hilbert ring with $|Max(T)|\leq 2^{\aleph_0}$ and $|T/J(T)|\leq 2^{2^{\aleph_0}}$. We observe that if $T$ is an algebraic $K$-algebra over a field $K$, then either $T$ has a maximal subring or $U(T)$ is integral over the prime subring of $T$. If $T$ is a left Artinian ring which is integral over its center, then we prove that either $T$ has a maximal subring or $T$ is countable and is integral over its prime subring. We see that if $T$ is a left Noetherian ring which is integral over its center, then either $T$ has a maximal subring or $|T|\leq 2^{\aleph_0}$. We prove that if $T$ is a domain which is integral over its center $C$ and $J(C)=0$, then either $T$ has a maximal subring or $T$ is an integral domain. If $T$ is a reduced ring which is integral over its center and the center of $T$ is a Hilbert ring, then we show that either $T$ has a maximal subring or $T$ is commutative. We see that if a ring $T$ is integral over its center and $R$ is a subring of $T$ with $J(T)\cap R\subseteq J(R)$, then either $T$ has a maximal subring or $J(R)=J(T)\cap R$ and $U(R)=U(T)\cap R$. Finally, we prove that if $T$ is direct product of an infinite family of rings $\{T_i\}_{i\in I}$ and each $T_i$ is integral over its center, then $T$ has a maximal subrings.