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**tttcomrader** Thanks for your massive help!!!

My last question here is: Prove that $\displaystyle I=J$ if and only if $\displaystyle IR_M=JR_M$, for each maximal ideal M of R.

Proof so far.

Suppose that I=J, then $\displaystyle IR_M = I = J = JR_M$, since I and J are ideals.

Conversely, suppose that $\displaystyle IR_M = JR_M$, pick an element in $\displaystyle IR_M$, then it has the form $\displaystyle i \frac {a}{b}, a \in R, b \in R \ not \ M, i \in I $.

Now, I=J implies that there exists an element $\displaystyle j \in J $ such that $\displaystyle j=i$, thus $\displaystyle i \frac {a}{b} = j \frac {a}{b} \in JR_M$, so we have $\displaystyle IR_M \subseteq JR_M$, and the reverse is the same.

Is this correct? My solution look a little bit too easy compare to the other problems...

Thanks!!!