The conductor ideals of maximal subrings in non-commutative rings

Abstract

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 $|Mi{n}_R((R:T))|\le 2$. We prove that if $T_R$ 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 $(R:T{)}_{l}T=T$, 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 $(R:T)\ne 0$. If $|Min(R)|\ge 3$, 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 $T$.