The Krull rings of Huckaba/Kennedy are defined in terms of their total quotient rings, while Lucas’s \(Q_0\) -Krull rings are defined in terms of their rings of finite fractions. Using the notion of a basis of regularity \(\mathcal {B}\) , we introduce \(Q_\mathcal {B}\) -Krull rings, which are defined in terms of their \(\mathcal {B}\) -quotient rings and specialize to Krull rings or \(Q_0\) -Krull rings for a suitable choice of \(\mathcal {B}\) . There are a plethora of well-known characterizations of Krull domains, involving a wide range of concepts such as t-invertibility, (complete) integral closure, prime factorization, and valuation rings. We generalize many of these classic theorems to \(Q_\mathcal {B}\) -Krull rings and prove several results that are new even in the Krull domain and/or ( \(Q_0\) -)Krull ring special cases.