Adding an ideal and then quotienting is the same as quotienting the subring by the intersection.Suggest a slogan
Let \(R\) be a ring, \(S \subseteq R\) a subring, and \(I \subseteq R\) a two-sided ideal. Then \(S + I\) is a subring of \(R\), \(S \intersect I\) is a two-sided ideal of \(S\), and there is a ring isomorphism \[ \frac{S+I}{I} \cong \frac{S}{S \intersect I}. \]