Dimension and Length for Artinian Modules

A local homology theory for linearly compact modules

“…Roberts [21] (see also [13]). Unfortunately, as in a personal communication of H. Zöschinger, he gave us the existence of semi-discrete linearly compact modules K of noetherian dimension 1 such that H m i (K) = 0 for all non-negative integers i.…”

“…Roberts[21] by the name Krull dimension. Later, D. Kirby[13] changed this terminology of Roberts and refereed to noetherian dimension to avoid confusion with well-known Krull dimension of finitely generated modules. Let M be an R-module.…”

On the Loewy Length and the Noetherian Dimension of Artinian Modules

“…Roberts [13], calls this dual dimension again Krull-dimension and applying the work of Kirby [8], on Hilbert polynomials for Artinian modules he proves that Artinian modules over commutative quasi-local rings have ®nite Noetherian-dimension. More recently, this dimension was called Noetherian-dimension by Kirby [9] and using this result of Roberts [13] and a method of sharp [14] and Matlis [12] (which reduces the study of Artinian modules over arbitrary commutative rings to study Artinian modules over quasi-local rings) he has proved that Artinian modules over commutative rings have ®nite Noetheriandimension.…”

On Α -QUASI SHORT MODULES

“…The dual Krull dimension in [7,8,11,12,13,14,15,16,17] is called Noetherian dimension and in [5] is called N-dimension. This dimension is called Krull dimension in [21].…”