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 \mr{pe.d}_A(M). This dimension measures how far a module is from torsion freeness. We show for every submodule \(N\) of \(M\), \(\mr{pe.d}_A(N)\leq\mr{pe.d}_A(M)\) and \(\mr{pe.d}_A(N)+\mr{pe.d}_A(M/N)\geq\mr{pe.d}_A(M)\). 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 \(M\).

Prime extension dimension of a module Keywords:

Prime submodules , Primary decomposition , Prime filtration and Regular prime extension filtration