## Divisor theories with primary elements and weakly Krull domains.(English)Zbl 0849.20041

We quote the author’s words: There are two different approaches towards classical arithmetic in algebraic number fields: the ideal-theoretic and the divisor-theoretic one. Both generalize to Krull domains. Whereas the divisor-theoretic approach stresses the purely multiplicative character of the arithmetic (a domain $$R$$ is a Krull domain if and only if its multiplicative monoid is a Krull monoid) it does up to now not generalize for non-normal domains (whereas ideal theory does). It is the aim of this paper to remove this deficiency for the class of weakly Krull domains as introduced and investigated elsewhere. According to a general philosophy, we do the main work in the context of monoids and prove that the corresponding concepts for domains are “purely multiplicative” (a domain $$R$$ is a weakly Krull domain if and only if its multiplicative monoid is a weakly Krull monoid).

### MSC:

 20M14 Commutative semigroups 13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) 13A05 Divisibility and factorizations in commutative rings 13A15 Ideals and multiplicative ideal theory in commutative rings 20M12 Ideal theory for semigroups