2024/09/12 by Alborz Azarang, Azarang, Alborz
Mathematics · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2410.10822
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)|≤ 2ℵ0 and |T/J(T)|≤ 2^2ℵ0. 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|≤ 2ℵ0. 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)∩ R⊆ J(R), then either T has a maximal subring or J(R)=J(T)∩ R and U(R)=U(T)∩ R. Finally, we prove that if T is direct product of an infinite family of rings \Ti\i∈ I and each Ti is integral over its center, then T has a maximal subrings.