# Prime extension dimension of a module

Document Type: Research Paper

Authors

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.

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 $M$, $\mr{pe.d}_A(N)\leq\mr{pe.d}_A(M)$ 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 $M$.

Keywords