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.