The conductor ideals of maximal subrings in non-commutative rings

Let R be a maximal subring of a ring T, and $(R:T)$, (R:T)ₗ and (R:T)ᵣ denote the greatest ideal, left ideal and right ideal of T which are contained in R, respectively. It is shown that (R:T)ₗ and (R:T)ᵣ are prime ideals of R and |MinR((R:T))|≤ 2. We prove that if TR has a maximal submodule, then (R:T)ₗ is a right primitive ideal of R. We investigate that when (R:T)ᵣ is a completely prime (right) ideal of R or T. If R is integrally closed in T, then (R:T)ₗ and (R:T)ᵣ are prime one-sided ideals of T. We observe that if (R:T)ₗT=T, then T is a finitely generated left R-module and (R:T)ₗ is a finitely generated right R-module. We prove that Char(R/(R:T)ₗ)=Char(R/(R:T)ᵣ), and if $Char(T)$ is neither zero or a prime number, then (R:T)≠ 0. If |Min(R)|≥ 3, then $(R:T)$ and (R:T)ₗ(R:T)ᵣ are nonzero ideals. Finally we study the Noetherian and the Artinian properties between R and T.