The theorem is stated in the context of commutative Banach (unitary) algebras, but the proof seems t

orlovskihmw 2022-07-11 Answered
The theorem is stated in the context of commutative Banach (unitary) algebras, but the proof seems to show that it is valid for any commutative algebra defined as a linear space where a commutative, associative and distributive (with respect to the addition) multiplication is defined such that α K α ( x y ) = ( α x ) y = x ( α y ).
In any case, whether it concerns only commutative Banach unitary algebras or commutative algebras as defined above, I think we must intend contained as properly contained. Am I right?
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Answers (1)

Nicolas Calhoun
Answered 2022-07-12 Author has 15 answers
in Lemma 2 only necessity is demonstrated, and the demonstration does no more than point to the general structure theorem for arbitrary commutative rings - that for a ring A and an ideal I the ideals over I in A are in 1-1 correspondence with the ideals of A / I.
the answer to your question is, therefore, yes.
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