ABSTRACT
CHAPTER 5
Let H be a reduced Krull monoid, F = F(P ) a monoid of divisors for H and G = F/H its class group (see Definition 2.4.9). For a ∈ F , let [a] = [a]F/H ∈ G be the class containing a, and denote by GP = {[p] | p ∈ P} the set of all classes containing primes. Let β˜ : F → F(GP ) be the class homomorphism, defined by β˜(p) = [p] for all p ∈ P , and let β = β˜ |H : H → B(GP ) be the block homomorphism of the Krull monoid H (see Definition 3.4.9). By Proposition 3.4.8, β is a transfer homomorphism. In particular, an element a ∈ H is an atom if and only if β(a) is an atom of B(GP ). But B(GP ) is a divisor-closed submonoid of B(G), and therefore an element of B(GP ) is an atom of B(GP ) if and only if it is an atom of B(G). Therefore the investigation of the structure of atoms in B(G) promotes our understanding of the factorizations in Krull monoids (see Theorem 5.1.5).
