Associated ideals to totally noetherian modules

Abstract

One problem in the study of the decomposition of modules is to choose the simple pieces to build such decompositions. In the noetherian case these simple pieces are the coprimary modules; therefore, each noetherian module is a subdirect product of finitely many coprimary modules, and each coprimary module has associated a unique prime ideal. A relative notion of noetherian modules was introduced by Anderson and Dumitrescu as S–noetherian modules, relative to a multiplicative subset S ⊆ A of a commutative ring, in [1]. Since then many authors have worked on this notion introducing prime and primary ideal and submodules relative to S. We have chosen a more general point of view, and work on a hereditary torsion theory σ in Mod–A and extend S–noetherian to totally σ–noetherian, recovering earlier notions when we take σ = σS. Since we first show that σ is of finite type whenever A is totally σ–noetherian, hence our theory can be regarded as an extension of the Anderson–Dumitrescu’s theory taking a multiplicative subset of finitely generated ideals instead of a multiplicative subset of elements. In this context we establish new results on prime and primary ideals and submodules, provide a primary decomposition of totally σ–noetherian modules, and show some applications of this primary decomposition, in particular, to totally σ–artinian modules.