### Finitely Generated Prime Ideals in H? and A (ID).

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A constructive proof of the Beurling-Rudin theorem on the characterization of the closed ideals in the disk algebra A(𝔻) is given.

We present an example of a subalgebra with infinite stable rank in the algebra of all bounded analytic functions in the unit disk.

It is shown how to embed the polydisk algebras (finite and infinite ones) into the disk algebra A(𝔻̅). As a consequence, one obtains uniform closed subalgebras of A(𝔻̅) which have arbitrarily prescribed stable ranks.

Using Baire's theorem, we give a very simple proof of a special version of the Lusin-Privalov theorem and deduce via Abel's theorem the Riemann-Cantor theorem on the uniqueness of the coefficients of pointwise convergent unilateral trigonometric series.

It is shown that the Bourgain algebra $A{\left(\right)}_{b}$ of the disk algebra A() with respect to ${H}^{\infty}\left(\right)$ is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to ${H}^{\infty}\left(\right)$, the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is $A{\left(\right)}_{b}$.

Let f be a function in the Douglas algebra A and let I be a finitely generated ideal in A. We give an estimate for the distance from f to I that allows us to generalize a result obtained by Bourgain for ${H}^{\infty}$ to arbitrary Douglas algebras.

We study the problem of simultaneous stabilization for the algebra ${A}_{\mathbb{R}}\left(\right)$. Invertible pairs $({f}_{j},{g}_{j})$, j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $\alpha {f}_{j}+\beta {g}_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since ${A}_{\mathbb{R}}\left(\right)$ has stable rank two, we are faced here with a different situation....

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