A Method for the Study of Artinian Modules, With an Application to Asymptotic Behavior

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.…”

ASYMPTOTIC STABILITY OF AttRTor1R((R/$Afr$n),A)

“…It follows (see [1]) that the sequence of sets Ass R (M/a n M ) is ultimately constant for large n. Assume A is an Artinian R-module. Dual to this result, Sharp has shown in [6,7] that the sequence of sets Att R (0 : A a n ) is ultimately constant for large n. Recently, in [3], Melkersson and Schenzel showed, in the case where R is Noetherian, that for each i the set of prime ideals Ass R Tor R i ((R/a n ), M) and Att R Ext i R ((R/a n ), A) become, for n large, independent of n. They also asked whether the sets Ass R Ext i R ((R/a n ), M) become stable for sufficiently large n. The aim of this note is to show that, for a finitely generated ideal a of R, the sequence of sets Att R Tor R 1 ((R/a n ), A) is ultimately constant for large n. This implies, under the Noetherian hypothesis on R, that the sequence of sets Ass R Ext 1 R ((R/a n ), M) become stable for sufficiently large n, which is an affirmative answer to the above question in the case i = 1.…”

On the Dual Notion of Multiplication Modules

“…Now each submodule of E = E(R/m) has a structure as R m (resp. R m ) module (see [14]). Further we have Ann R (N )R m = Ann R m (N ).…”