Short modules and almost noetherian modules

Abstract

It is proved that, for any ring R, a right R-module M has the property that, for every submodule N, either N or M/N is Noetherian if and only if M contains submodules K ⊇ L such that M/K and L are Noetherian and K/L is almost Noetherian. Short modules and almost Noetherian modulesAll rings are associative with identity and all modules are unital right modules. Let R be any ring. In [11], Sarath defines an R-module M to be tall if M contains a submodule N such that N and M/N are both non-Noetherian. We shall call an R-module short if it is not tall. Thus a module M is short if and only if, for each submodule N of M, either N or M/N is Noetherian. Clearly every Noetherian module is short. As we shall see below, it is easy to produce examples of short modules which are not Noetherian.Following [2], we call an R-module M almost Noetherian if every proper submodule of M is finitely generated. Clearly a module M is almost Noetherian if every proper submodule of M is Noetherian. Clearly also, almost Noetherian modules are short. It is proved in [6, Theorem 2.2] that if Z is the ring of rational integers then a Z-module M is almost Noetherian if and only if M is Noetherian or is isomorphic to the Prüfer p-group Z(p ∞ ) for some prime p. In [2, Theorem 2.1], Armenderiz characterized all commutative rings R such that the ring of fractions of R is an almost Noetherian R-module. In particular, if R is a discrete valuation ring then the field of fractions K of R is an almost Noetherian R-module, so that K is a short R-module which is not Noetherian.In [13], an R-module M is called almost finitely generated (a.f.g.) if M is not finitely generated as an R-module but every proper R-submodule of M is finitely generated. (Note that in [5], a.f.g. modules are called "Jónsson w 0 -generated modules".) Weakley [13] proved that if R is a commutative ring and M is an a.f.g. R-module then P = {r ∈ R : rM = 0} is a prime ideal of R.

Max Modules Relative to a Torsion Theory

“…(1) Suppose that M is almost τ -Noetherian and let K ≤ N ≤ M be such that N ∈ D τ (M ). Then K ∈ D τ (M ) and so K is τ -finitely generated and thus, by Proposition 5.3 (4), N is τ -Noetherian. Conversely if N ∈ D τ (M ) then it is τ -Noetherian and thus τ -finitely generated.…”

“…We now pave the way to our characterization of τ -short modules by a series of results following those used in the proof of the characterization of short modules in [4].…”

“…In this section we introduce τ -short modules motivated by a recent article by Bilhan and Smith [4] which defined short modules and characterized them using Shock's Theorem. We follow a similar line of approach culminating in a torsion-theoretic analogue of their characterization which relies on Theorem 5.12, our analogue of Shock's Theorem.…”

“…The first is derived from [9] where a module M is called tall if there is an N ≤ M such that both N and M/N are non-Noetherian. The next is adapted from [4] where M is called short if it is not tall. The last comes from [2] where M is called almost Noetherian if every proper submodule of M is finitely generated.…”

On Α -QUASI SHORT MODULES

“…Bilhan and Smith have introduced and extensively investigated short modules and almost Noetherian modules, see [4]. Later Davoudian, Karamzadeh and Shirali undertook a systematic study of the concepts of α-short modules and α-almost Noetherian modules, see [8].…”