Prime extension dimension of a module

Document Type : Research Paper


1 Department of Mathematics, Pondicherry University, Puducherry, India.

2 Department of Mathematics, Pondicherry Engineering College, Puducherry, India.

3 Department of Mathematics, Indira Gandhi Institute of Technology, Odisha, India.


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 \(M\).