PRIME EXTENSION DIMENSION OF A MODULE

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 pe.dA(M). This dimension measures how far a module is from torsion freeness. We show for every submodule N of M, pe.dA(N) ≤ pe.dA(M) and pe.dA(N)+pe.dA(M/N) ≥ pe.dA(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.