Second isomorphism theorem for rings

Second isomorphism theorem (rings): Let $R$ be a ring, let $A\subseteq R$ be a subring , and let $I\triangleleft R$ be an ideal . Then $A\cap I\triangleleft A$, $A+I:=\{a+i:a\in A,\ i\in I\}$ is a subring of $R$, and there is a natural isomorphism

of quotient rings .

This is most efficiently proved by restricting the quotient map $R\to R/I$ to $A$ and applying the first isomorphism theorem .