Classical ideals theory of maximal subrings in non-commutative rings

Let $R$ be a maximal subring of a ring $T$. In this paper we study relation between some important ideals in the ring extension $R\subseteq T$. In fact, we would like to find some relation between $Nil_*(R)$ and $Nil_*(T)$, $Nil^*(R)$ and $Nil^*(T)$, $J(R)$ and $J(T)$, $Soc({}_RR)$ and $Soc({}_RT)$, and finally $Z({}_RR)$ and $Z({}_RT)$; especially, in certain cases, for example when $T$ is a reduced ring, $R$ (or $T$) is a left Artinian ring, or $R$ is a certain maximal subring of $T$. We show that either $Soc({}_RR)=Soc({}_RT)$ or $(R:T)_r$ (the greatest right ideal of $T$ which is contained in $R$) is a left primitive ideal of $R$. We prove that if $T$ is a reduced ring, then either $Z({}_RT)=0$ or $Z({}_RT)$ is a minimal ideal of $T$, $T=R\oplus Z({}_RT)$, and $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. If $T=R\oplus I$, where $I$ is an ideal of $T$, then we completely determine relation between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of $R$ and $T$. Finally, we study the relation between previous ideals of $R$ and $T$ when either $R$ or $T$ is a left Artinian ring.