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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.

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