I am not quite sure the license has been activated. how can I check it?
If the node renders when you send it a job, it's activated. :) The devs probably have a better answer, but I don't think Corona is technically activated until it tries to render and looks for a license. The batch file you run just tells it where to look for the license when it asks for it. So it won't be active until you render with it once.
furthermore when I have to create the .bat file exactley where do I have to save it? there is no location specified..
I just keep the .bat file on our network next to the installer file. I don't have an automatic deployment set up yet, so to install Corona I remote in to the render node, run the installer, then run the batch file.
and one more thing, do i have to run the bat file everytime or just once and then I can remove the file?
You just run the batch file once. The activation instructions remain even if you upgrade Corona to a new version.

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i9-12900K @ 3.2GHz, 64GB RAM, 3090ti
Max 2025, Corona 12