The monotone catenary degree of monoids of ideals

Abstract

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of vinvertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.has factorizations with distance greater than M . However, at least those factorizations, which are powers of factorizations of a, can be concatenated, step by step, by factorizations whose distance is small and does not depend on M . This phenomenon is formalized by the catenary degree which is defined as follows. The catenary degree c(H) of H is the smallest N ∈ N 0 ∪ {∞} such that for each a ∈ H and each two factorizations z, z ′ of a there is a concatenating chain of factorizations z = z 0 , z 1 , . . . , z k+1 = z ′ of a such that the distance d(z i−1 , z i ) between two successive factorizations is bounded by N . It is well-known that the catenary degree is finite for Krull monoids with finite class group and for C-monoids (these include Mori domains R with nonzero conductor f = (R : R) for which the residue class ring R/f and the class group C( R) are finite).In order to study further structural properties of concatenating chains, Foroutan introduced the monotone catenary degree ([13]). The monotone catenary degree c mon (H) is the smallest N ∈ N 0 ∪ {∞} such that for each a ∈ H and each two factorizations z, z ′ of a there is a concatenating chain of factorizations z = z 0 , z 1 , . . . , z k+1 = z ′ of a such that the distance between two successive factorizations is bounded by N and in addition the sequence of lengths |z 0 |, . . . , |z k+1 | of the factorizations is monotone (thus either |z 0 | ≤ . . . ≤ |z k+1 | or |z 0 | ≥ . . . ≥ |z k+1 |). Therefore, by definition, we have c(H) ≤ c mon (H) and2010 Mathematics Subject Classification. 13A15, 13F15, 20M12, 20M13.

On the arithmetic of stable domains

“…For the following equivalent statements, let r be an ideal system on R such that every r -ideal of R is an ideal of R . We say that R is a Cohen-Kaplansky domain if one of the following equivalent statements hold [ 4 , Theorem 4.3] and [ 25 , Proposition 4.5]. R is atomic and has only finitely many atoms up to associates.…”

“…The concept of strongly ring-like monoids was introduced by Hassler [ 35 ], and the question which one-dimensional local domains are strongly ring-like was studied in [ 25 , Section 5].…”

“…By [ 21 , Proposition 2.10.7], one-dimensional local Mori domains with non-zero conductor are finitely primary of rank By Corollary 3.2.2 , is a Dedekind domain with at most two maximal ideals, whence Since every ideal of R is 2-generated by Proposition 3.5.4, whence R is noetherian. If is the maximal ideal of R , then and whence is strongly ring-like by [ 25 , Corollary 5.7]. If s = 2, then by (5.4).…”

“…By Proposition 5.2, the localizations R p are strongly ring-like of rank at most two. Thus I Ã ðRÞ has finite monotone catenary degree by[25, Theorem 5.13].3. By[21, Proposition 1.4.5] and by Proposition 4.3, we have qðI Ã ðRÞÞ ¼ supfqðR p Þ j p 2 XðRÞg :…”

For Which Puiseux Monoids Are Their Monoid Rings Over Fields Ap?

“…(Note that the notions of an atomic and an ACCP monoid are analogous to the notions of an atomic and an ACCP integral domain. The similarities and differences between the ideal theories of monoids and integral domains are studied, for example, in the classical references [3,29,31], as well as in the recent papers [15,16,17]. )…”