The first thing to do is open the dictionary and look up the word, "friend." You are actually talking about an acquaintance, and even that is a stretch. Before your manager commissions a sculptor to make a statue of her, you need to sit down with her and have a little chat.
The best approach is to use a low-key, assertive, and businesslike style. Your opening comments should focus on her specific behavior that you found to be so questionable -- namely, her taking your idea to management. The idea was yours and she had no right to do this. The next step is to tell her how you feel about her actions. Let her know that this was upsetting and stressful, and you feel hurt and betrayed.
Having set this groundwork, you should then suggest that the two of you meet with the manager to discuss the matter. Let her know that both of you will look better in the manager's eyes if you sit down with him and honestly explain what happened.
If she categorically refuses to do this, tell her that you plan to approach the manager on your own. Incidentally, if you happen to have any kind of proof that the idea was yours, let her know that you will be bringing it with you. In your meeting with the manager, one effective approach is to tell him that you have a problem at work and you would like his help in solving it. Then describe the entire incident. If he is any kind of a manager, he will start his investigation immediately.
The larger question for you is whether you are afraid to act assertively. Being assertive is not being aggressive or obnoxious; it is merely standing up for your rights and beliefs. If you don't do so, it is quite likely that there will be others who steal ideas and opportunities that are rightfully yours.
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