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

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 $\mathrm{\forall }\alpha \in \mathbb{K}\phantom{\rule{1em}{0ex}}\alpha \left(xy\right)=\left(\alpha x\right)y=x\left(\alpha y\right)$.
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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Nicolas Calhoun
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 $A/I$.