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This could easily be misunderstood as saying the disk algebra is the algebra of holomorphic functions on the open unit disk AND that every such function can be extended to a continuous complex-valued function on the closure. But that is false, as shown by the example
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by a single element. But I think that to be worth stating this should be given in the context of a description of the ideals (to be found for example in
Chapter 6 of Hoffman's book). I'm not interested in doing this anytime soon.
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I want to clarify an edit I made. I removed the statement that the disk algebra is a PID because I thought it was misleading without context. An alternative would be to provide the context and state precisely what is meant. Every
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