Non-commutative rings with infinitely many maximal subrings

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.