Second isomorphism theorem for rings

Statement

Let $R$ be a ring, let $S$ be a subring of $R$ and let $I$ be a two-sided ideal of $R$. Then $S + I = \{s + x : s \in S, x \in I\}$ is a subring of $R$, and $I$ is a two-sided ideal of $S + I$. The intersection $S \cap I$ is a two-sided ideal of $S$. The map $S/(S \cap I) \to (S + I)/I$, $s + (S \cap I) \mapsto s + I$, is a ring isomorphism.

Remarks

The proof applies the first isomorphism theorem to the map $S \to (S + I)/I$, $s \mapsto s + I$. It is surjective, and its kernel is $S \cap I$. The theorem is also called the diamond theorem, after the picture of the four rings $S \cap I$, $S$, $I$ and $S + I$.

Explanations

L1 human

The theorem compares two quotients. Divide the subring $S$ by its part inside $I$, or divide the larger ring $S + I$ by $I$. The two quotients are the same ring.