Fitting ideals and the Gorenstein property

TL;DR

This paper establishes an inequality relating the size of finite modules over group rings to their Fitting ideals, under specific conditions on the group order and prime, contributing to the understanding of module structure in algebra.

Contribution

It proves a new inequality linking module size and Fitting ideals over group rings when the group order condition is met.

Findings

The inequality #M ≤ #Z_{p}[G]/Fit_{Z_{p}[G]}(M) holds under the given conditions.

The result applies to modules over group rings with prime restrictions.

Provides insight into the structure of modules over group rings in algebra.

Abstract

Let p be a prime number and G be a finite commutative group such that p^{2} does not divide the order of G. In this note we prove that for every finite module M over the group ring Z_{p}[G], the inequality #M \leq #Z_{p}[G]/Fit_{Z_{p}[G]}(M) holds. Here, Fit_{Z_{p}[G]}(M) is the Z_{p}[G]-Fitting ideal of M.