Ring (not Algebra) Isomorphisms of H(G)

Abstract

We return here to the ring structure of H(G). A ring homomorphism of R₁ to R₂ is a function φ: R₁ → R₂ which preserves multiplication i.e. φ(rs) = φ(r)φ(s) and φ(r + s) = φ(r) + φ(s) for all r, s ∈ R₁. A ring isomorphism is a ring homomorphism that is one-to-one and onto. If G and G’ are two conformally equivalent domains in ℂ then there is an algebra isomorphism from H(G) to H(G’) as we saw in Chapter 5. An algebra isomorphism will be a ring isomorphism which additionally preserves scalar multiplication. It follows from Proposition 5.4 that H(G) and H(G’) are isomorphic as algebras if and only if G and G’ are conformally equivalent. But a ring isomorphism can exist without conformal equivalence.