Maximal Ideal Theorem: Proving M/IM=O for R-Module M and Ideal I

[mclove]

Homework Statement

let M be an R-module and I an ideal of R
suppose Mm=0 for all maximal ideals m
then, M=IM

The Attempt at a Solution

O\rightarrow IM \rightarrow M \rightarrow M/IM \rightarrow O is exact seq
To show that M/IM=O (then M=IM)

 

[Petek]

Is M a left R-module or a right R-module? You have R operating both on the right (Mm = 0) and the left (M = IM) of M. Also, assuming that all R operations are on the left (say), then what if I is a maximal ideal? We'd then have that M = MI = 0, which doesn't seem correct.

Petek