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⊆ 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⊕ Z(RT), and (R:T)=(R:T)ₗ=(R:T)ᵣ=annR(Z(RT)). If T=R⊕ 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.