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.