Maximal Ideal iff Quotient Ring is Field

Theorem
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $J$ be an ideal of $R$.

The following are equivalent:


 * $(1): \quad$ $J$ is a maximal ideal.


 * $(2): \quad$ The quotient ring $R / J$ is a field.

Also see

 * Maximal Left and Right Ideal iff Quotient Ring is Division Ring