## Weakly Krull Inside Factorial Domains

Chapman, Halter-Koch, and Krause [8] introduced the notion of an inside factorial monoid and integral domain. Throughout we will conﬁne ourselves to the integral domain case, but the interested reader may easily supply deﬁnitions and proofs for the monoid case. An integral domain D is inside factorial if there exists a divisor homomorphism ϕ: F → D∗ = D − {0} where F is a factorial monoid and for each x ∈ D∗, there exists an n ≥ 1 with xn ∈ ϕ(F ). They showed [8, Proposition 4] that an inside factorial domain may be deﬁned in terms of a Cale basis and it will be this characterization that we use for the deﬁnition of an inside factorial domain. A subset Q ⊆ D∗ is a Cale basis for D if 〈Q〉 = {uqα1 · · · qαn | u ∈ U(D), qαi ∈ Q} is a factorial monoid with primes Q and for each d ∈ D∗ there exists an n ≥ 1 with dn ∈ 〈Q〉. Here U(D) is the group of units of D. A domain D is inside factorial if and only if D has a Cale basis. They showed [8, Theorem 4] that D is inside factorial if and only if D, the integral closure of D, is a generalized Krull domain with torsion t-class group Clt(D), for each P ∈ X(1)(D), the valuation domain DP has value group order-isomorphic to a subgroup of (Q,+) (we say that D is rational), and D ⊆ D is a root extension (i.e., for each x ∈ D, there exists an n ≥ 1 with xn ∈ D).