Non-commutative rings with infinitely many maximal subrings

TL;DR

This paper investigates the structure and quantity of maximal subrings in various classes of rings, establishing conditions under which rings have finitely or infinitely many maximal subrings, with detailed classifications for specific ring types.

Contribution

It provides new criteria for the finiteness of maximal subrings in rings, especially in relation to algebraic properties, centers, and ring decompositions, extending previous understanding.

Findings

Rings with a maximal ideal not an ideal have at least |R/M|+1 maximal subrings.

If T is a K-algebra over an infinite field, it either has infinitely many maximal subrings or is a quasi duo ring.

For simple rings R, R×R has finitely many maximal subrings iff R is finite.

Abstract

We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal left/right ideals which are not an ideal of $T$; in particular $T$ has at least $|R/M|+1$ distinct maximal subrings. Moreover, if $T$ is a $K$-algebra over an infinite field $K$, then either $T$ has infinitely many maximal subrings or $T$ is a quasi duo ring with certain algebraic properties similar to commutative rings. We prove that for a simple ring $R$, the ring $R\times R$ has only finitely many maximal subrings if and only if $R$ is finite. Also we study rings which are integral over their centers and have only finitely many maximal subrings. We prove that if $T$ is integral over its center and $T$ has more than $2^{\aleph_0}$ maximal (left/right) ideals, then $T$ has infinitely many maximal subrings. In particular, we see that if a $J$-semisimple ring $T$ is integral over its center and has only finitely many maximal subrings, then $T$ embeds in $S\times \prod_{i\in I}E_i$, where each $E_i$ is an absolutely algebraic field and $S$ is a finite semisimple ring. We see that if $T$ is a left Noetherian algebraic $K$-algebra over an infinite field $K$ and $T$ has only finitely many maximal subrings, then $T$ is a countable left Artinian ring which is integral over $\mathbb{Z}_p$, where $p=Char(K)$. We exactly determine when $T=\prod_{i\in I}\mathbb{M}_{n_i}(E_i)$, where each $E_i$ is a field and $n_i\in\mathbb{N}$, has only finitely many maximal subrings. We see that if $R$ is an infinite Artinian ring, then $\mathbb{M}_n(R)$, $n>1$, and $R\times R$ have infinitely many maximal subrings.