10 points each Problem S. Let R = ( R, *) be a commutative ring with unity , and suppose that M is a maximal proper ideal of R . In this exercise , we will prove that M must be prime . Suppose by way of contradiction that there exist a , b ER - M such that a * b E M . Part ( a) . In the homework for Investigation 22 , you showed that My = (m + U* X : X ER ; is an ideal that contains the set M for any U ER. Use this fact and our assumption to explain why we know there* exist X , Y ER and m , n2 EM such that a* * + m = 12 - 6 * } + 12. Part ( b ) . Use the fact that IR = 1R* IR to prove that IR EM _ _ _ contrary to assumption