User:J D Bowen/Math735 HW5

Since $$M \ $$ is a maximal ideal, it is prime. Let $$x,y\in R-M $$. If we had $$xy\in M \ $$, then since $$M \ $$ is prime, we have $$x\in M \ $$ or $$y \in M \ $$. Hence, $$R-M \ $$ is closed under multiplication.