Roddomänen

Roddomänen, also called rod domains, are a proposed class of commutative integral domains defined by a hierarchical decomposition of their multiplicative structure into a sequence of subrings called rods. A rod-domain R is equipped with an increasing filtration of subrings (Rα)α∈Ord such that R0 ⊆ R1 ⊆ ... ⊆ ⋃α Rα = R. Each successive rod captures a distinct layer of divisibility. A nonzero element a ∈ R is factored as a = u r1 r2 ... rn, where u is a unit and each ri lies in a later rod than the previous one, with the indices strictly increasing. The number n, called the rod-length of a and denoted ℓ(a), measures the depth of a’s decomposition.

Roddomänen generalize several classical domain classes. If the filtration has length one, the rod-domain reduces to a principal ideal domain. If the filtration aligns with a valuation, one recovers a valuation domain. The framework links factorization properties with graded and hierarchical structures, introducing invariants such as rod-length and rod-graded rank that enrich the study of divisibility in non-Noetherian settings.

Examples and constructions. Rank-two valuation domains admit natural rod-filtrations, producing elementary rod-domains. More complex instances arise by equipping polynomial or power-series rings with lexicographic filtrations on monomials, yielding multi-layer rod-domains.

Applications. Rod-domains appear in the investigation of graded algebras, in factorization questions beyond Noetherian assumptions, and in algebraic-geometric contexts where hierarchical valuations reflect geometric filtrations or filtrations of coordinate rings.

See also. Integral domain, valuation domain, principal ideal domain, unique factorization domain, graded ring.