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.