Homogeneous Weights and M\"obius Functions on Finite Rings

TL;DR

This paper explores homogeneous weights, M"obius functions, and Euler phi-functions on finite rings, providing computational formulas and linking them to classical number theory functions for residue rings of integers.

Contribution

It introduces computational formulas for these functions on finite principal ideal rings and connects them to classical number-theoretic functions.

Findings

Derived formulas for functions on finite principal ideal rings

Reduced functions to classical number-theoretic M"obius and Euler phi-functions for residue rings

Characterized properties of these functions on finite rings

Abstract

The homogeneous weights and the M\"obius functions and Euler phi-functions on finite rings are discussed; some computational formulas for these functions on finite principal ideal rings are characterized; for the residue rings of integers, they are reduced to the classical number-theoretical M\"obius functions and the classical number-theoretical Euler phi-functions.