Quasi-divisor theories and generalizations of Krull domains

Ordered groups with greatest common divisors theory

“…Then according to previous lemma every finitely generated v-ideal of G = G(R) is v-invertible, but finitely generated v-ideal of G do not form a group. Hence, G does not admit quasi divisor theory according to [2] or [7]. On the other hand, G admits GCD theory as the following proposition shows (a version of this proposition for integral domains was proved by Lucius [10]).…”

“…This mistake was pointed out by Lucius [10] who proved (in language of integral domains) that (1.3) does not imply (1.1), in general, and he introduced (again in the language of integral domains) a new notion of a po-group with the GCD theory as a po-group G with the embedding h : G / / Γ which satisfies the condition (1.3). He proved also that the group of divisibility of an integral domain R satisfies the condition (1.3) if and only if R is v-domain on the contrary to the result of Geroldinger and Močkoř [7] stating that the group of divisibility of R satisfies the condition (1.1) if and only if R is a Prüfer v-multiplication domain. Hence, po-groups which satisfy the condition (1.3) really represent a generalization of po-groups with quasi divisor theory.…”

“…After then this axiomatic concept was firstly transformed onto semigroups with cancellation by Skula [16,17] and then transformed and generalized onto po-groups and monoids by several authors (see [2,6,7,8,13,14]). Nevertheless in all these transformations and generalizations the principal structure of this axiomatic concept remains the same and for po-groups it could be expressed as follows.…”

“…(1.4) In po-groups and integral domains theory these conditions (1.1), (1.2), (1.3), and (1.4) characterize po-groups with various types of divisor theory or integral domains which are generalizations of Krull domains (see [2,7,13,16]). For example, if Γ is required to be a free abelian group Z (P ) for some set P then the conditions (1.1), (1.2), and (1.3) are equivalent and h is then called the theory of divisors of G. For general l-group Γ the condition (1.1) characterizes po-groups with the so-called theory of quasi divisors (this notion was introduced by Aubert [2], although Jaffard investigated properties of such groups more early [9]), and the condition (1.4) leads to po-groups with the strong theory of quasi divisors (see [14]).…”

“…There exists a lot of works justifying these processes, one of the last result in this direction being a paper of Geroldinger and the author (see [7]) where it is proved that properties of being a Prüfer v-multiplication domain, a domain of Krull type, or an independent domain of Krull type, are purely multiplicative ones and can be expressed by using r -ideals in the corresponding groups of divisibility.…”