[Math] Prove that a proper ideal of R is a maximal ideal of R, with R being a commutative ring with unity

Question

R is a commutative ring with unity and A is a proper ideal of R, and every element of R that is not in A is a unit of R. Prove that A is a maximal ideal of R.

Can anyone give a head start on how to approach this problem? I only know I'm supposed to find a B=A or B=R with $A\subseteq B\subseteq R$.

Answer 1

Hint: If $B \supsetneq A$, $B$ contains a unit. By absorption, show that $1 \in B$. By absorption again, show $R \subseteq B$.