The Bézout properties for some classical function algebras

We prove that for infinite compact planar sets K with big complementary components the algebras P(K),R(K),A(K) (and C(K)) as well as the Sarason algebra H∞+C on the unit circle are quasi pre-Bézout rings that do not have the Bézout property. It is also shown that for a compact Hausdorff space X the real algebra C(X,τ) has the pre-Bézout property, but that surprisingly, C(X,τ) may be a Bézout-ring without X being an F-space. We also present several classes of rings of holomorphic functions in several complex variables that do not have the Bézout property.