Noetherian and totally noetherian rings and modules

Abstract

Given a commutative ring A there are different approaches to understand its structure; one is consider ideals and their arithmetic (multiplicative theory), and another one is to consider modules over A (module theory); in this work we shall mix both; on one hand we shall study ideals; in particular prime ideals, and on the other we shall use categories of modules and functors between them. Recall that the spectrum of A, endowed with Zariski topology, is a bridge between Algebra and Geometry. In this approach we shall consider subsets of the spectrum of A and chain conditions and presheaf constructions on them. Indeed, given a ring Awe shall consider a subsetK ⊆ Spec(A) closed under generalizations, and the associated hereditary torsion theory σK , or, more generally we shall consider a hereditary torsion theory σ on Mod–A, and define chain conditions relative to σ such that we extend the range of examples we may study. The behaviour of these constructions is acceptable from a categorical point of view, so we can construct new categories and functors and so on. The simplest example is provided by a multiplicative set S ⊆ A for which we have the fraction ring S−1A, and the category of S−1A modules. A general hereditary torsion theory σ has a similar description whenever A is a σ–noetherian ring; in fact, it is determined by a multiplicative set of finitely generated ideals rather than principal ones. In both cases we obtain a categorical framework which is useful for some developments; however, a more arithmetic approach might be of interest. For instance, an A–module M is σ–torsion when for each element m ∈ M there exists an ideal hm ∈ L(σ) such that mhm = 0; a common ideal h ∈ L(σ) should be the best choice to work more effectively; this occurs when M is finitely generated; that is, if M is σ–torsion and finitely generated, there exists an ideal h ∈ L(σ) such that Mh = 0. With this new approach we have three different notions for noetherian module: • M is noetherian whenever the lattice of all submodules of M is noetherian. • M is σ–noetherian whenever the lattice of all σ–closed submodules is noetherian, and the third one for which we have no categorical description is: • M is totally σ–noetherian whenever for any ascending chain of submodules {Ni | i ∈ I} there is an ideal h ∈ L(σ), and an element j ∈ I such that Nih ⊆ Nj . This notion of totally σ–noetherian was introduced by Anderson and Dumitrescu as S–finite for a multiplicative set S ⊆ A, which coincides with our definition of totally σS–noetherian. This more arithmetical approach to chain conditions has the advantage of allowing effective computation, and the disadvantage of losing several categorical and functorial constructions.