Correspondence Theorem for Quotient Rings

Theorem
Let $A$ be a commutative ring with unity.

Let $\mathfrak a \subseteq A$ be an ideal.

Let $A / \mathfrak a$ be the quotient ring and $\pi : A \to A / \mathfrak a$ the quotient ring epimorphism.

Also see

 * First Isomorphism Theorem for Rings
 * Universal Property of Quotient Ring
 * Correspondence Theorem for Localizations of Rings

Generalization

 * Correspondence Theorem for Surjective Ring Homomorphisms