Direct Report Who I Cannot Manage

Hi Coaches,

I have a story about my co-worker who is also my direct report. I want to be collaborative with my team, but whenever I share an idea that I want to introduce within the firm, she goes behind my back and steals the idea. E.g. Yesterday I shared with her and another associate that I was planning to research ways that our company can create videos and live events more efficiently. She immediately emailed our CEO after our meeting that she was interested in doing that and now is spearheading the initiative as if it were her own.

Although I want to be collaborative, my thoughts about her are causing me to feel defensive and protective, with my actions being to not share with her my ideas. I feel like this would give me an opportunity to lead the projects that I want, but on the other hand, it makes me less of the manager that I would like to be. How can I see her as a collaborative team member when I cannot control her behavior?