In this paper we obtain relations between some important ideals in the ring extension \(R\subseteq T\) , where R is a maximal subring of a ring T. In fact, we find some relations 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 \(Z({}_RT)\) . Special attention is given to cases such as when T is a reduced ring or whenever R (or T) is a left Artinian ring. If R is a maximal subring of a ring T, then we show that either \(Soc({}_RR)=Soc({}_RT)\) or \((R:T)_r\) (the greatest right ideal of T 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 \((R:T)=(R:T)_\ell =(R:T)_r=ann_R(Z({}_RT))\) . If \(T=R\oplus I\) , where I is an ideal of T, then we completely determine relations between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of R and T, in particular whenever R or T is a left Artinian ring.
