  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 $|Min_R((R:T))|\leq 2$. 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 $(R:T)_lT=T$, 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 $(R:T)\neq 0$. If $|Min(R)|\geq 3$,
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 $T$.
