Maximal subrings of certain noncommutative rings

The existence of maximal subrings in certain noncommutative rings, especially in rings which are integral over their centers, are investigated. We prove that if a ring [Formula: see text] is integral over its center, then either [Formula: see text] has a maximal subring or [Formula: see text] is a commutative Hilbert ring with [Formula: see text] and [Formula: see text]. We observe that if [Formula: see text] is an algebraic [Formula: see text]-algebra over a field [Formula: see text], then either [Formula: see text] has a maximal subring or [Formula: see text] is integral over the prime subring of [Formula: see text]. If [Formula: see text] is a left Artinian ring which is integral over its center, then we prove that either [Formula: see text] has a maximal subring or [Formula: see text] is countable and is integral over its prime subring. We see that if [Formula: see text] is a left Noetherian ring which is integral over its center, then either [Formula: see text] has a maximal subring or [Formula: see text]. We prove that if [Formula: see text] is a domain which is integral over its center [Formula: see text] and [Formula: see text], then either [Formula: see text] has a maximal subring or [Formula: see text] is an integral domain. If [Formula: see text] is a reduced ring which is integral over its center and the center of [Formula: see text] is a Hilbert ring, then we show that either [Formula: see text] has a maximal subring or [Formula: see text] is commutative. We see that if a ring [Formula: see text] is integral over its center and [Formula: see text] is a subring of [Formula: see text] with [Formula: see text], then either [Formula: see text] has a maximal subring or [Formula: see text] and [Formula: see text]. Finally, we prove that if [Formula: see text] is direct product of an infinite family of rings [Formula: see text] and each [Formula: see text] is integral over its center, then [Formula: see text] has a maximal subrings.