Any idea why my battery light on my dash stays after turning off car and removing the key?

If a diode goes out it allows current to flow backwards through the system. This is the same type of diode used in solar panels to keep the current from coming out of the batteries during periods of darkness. Sometimes called a "blocking diode", "Back out diode". Unless you are good at board work you will need to replace the whole diode pack.

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"power rectifying diode"

the quick solution is replace the alternator. good luck, chuck.