Arithmetic modules over generalized Dedekind domains

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

Resolving Decompositions for Polynomial Modules

We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.

Attached prime ideals of generalized inverse polynomial modules

Let [Formula: see text] be a ring, [Formula: see text] a strictly totally ordered monoid which is also artinian and [Formula: see text] a monoid homomorphism. Given a right [Formula: see text]-module [Formula: see text], denote by [Formula: see text] the generalized inverse polynomial module over the skew generalized power series ring [Formula: see text]. It is shown in this paper that if [Formula: see text] is a completely [Formula: see text]-compatible module and [Formula: see text] an attached prime ideal of [Formula: see text], then [Formula: see text] is an attached prime ideal of [Formula: see text], and that if [Formula: see text] is a completely [Formula: see text]-compatible Bass module, then every attached prime ideal of [Formula: see text] can be written as the form of [Formula: see text] where [Formula: see text] is an attached prime ideal of [Formula: see text].

ON LCM-STABLE MODULES

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.

ASSOCIATED PRIMES OVER ORE EXTENSION RINGS

The study of the prime ideals in Ore extension rings R[x,σ,δ] has attracted a lot of attention in recent years and has proven to be a challenging undertaking ([5], [7], [12], et al.). The present article makes a contribution to this study for the associated prime ideals. More precisely, we aim to describe how the associated primes of an R-module MR behave under passage to the polynomial module M[x] over an Ore extension R[x,σ,δ]. If we impose natural σ-compatibility and δ-compatibility assumptions on the module MR (see Sec. 2 below), we can describe all associated primes of the R[x,σ,δ]-module M[x] in terms of the associated primes of MR in a very straightforward way. This result generalizes the author's recent work [1] on skew polynomial rings.

Polynomial Modules Over Macaulay Modules

In [2] we introduced a concept of a Macaulay module over a right noetherian ring by saying that all associated primes of the module have the same codimension. That is to say that a module M over a right noetherian ring R is Macaulay if K dim R/P = K dim R/Q for all P, Q ∈ Ass M. Our main aim here is to extend Nagata’s useful result [6], that Macaulay rings are stable under polynomial adjunction, to a noncommutative setting. Specifically, we prove where x is a commuting indeterminate, that the polynomial module M[x] = M⊗RR[x] is a Macaulay R[x]-module if and only if M is a Macaulay R-module. But actually, we prove a more general result. We show that when M is any module over a right noetherian ring, the associated primes of M[x] are precisely the extensions of the associated primes of M.