No, Ujoe, the Flame of Arnor is Gandalf's power. He is a servant of the secret fire, and a weilder of the Flame of Arnor. He both serves and weilds the power. The ring of Gandalf was Narya, which was not his true power as a miar. It is speculated the Flame of Arnor is an allusion to the white light of the sun as a symbol of the secret fire. While it is no means certain, they are generally used as the same thing.
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