Maximal subrings of certain non-commutative rings
Alborz Azarang
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.