Maximal Ideal iff Quotient Ring is Field/Proof 1

Proof Outline
The hard part is proving the existence of inverses.

Take an element $x$ of the Ring to invert.

Define a set $K$, that contains the Ideal and is contained by the ring.

It is the set of all members of the Ideal, each added to a multiple of $x$.

Prove that this set is an ideal and contains our original maximal ideal.