Prime extension dimension of a module

Abstract

We have that for a finitely generated module $M$ over a Noetherian ring $A$ any two RPE filtrations of $M$ have same length. We call this length as prime extension dimension of $M$ and denote it as $mr{pe.d}_A(M)$. This dimension measures how far a module is from torsion freeness. We show for every submodule (N) of (M), (mr{pe.d}_A(N)leqmr{pe.d}_A(M)) and (mr{pe.d}_A(N)+mr{pe.d}_A(M/N)geqmr{pe.d}_A(M)). We compute the prime extension dimension of a module using the prime extension dimensions of its primary submodules which occurs in a minimal primary decomposition of (0) in (M).