An additional structure over integer rings $\mathbb{Z}_{p^r}^n$

TL;DR

This paper introduces a new algebraic structure over modules of integer rings with prime power cardinality, enabling basis definitions and extending linear algebra concepts to these modules.

Contribution

It develops an algebraic framework over integer rings of prime power order, extending basis theory and duality results beyond fields.

Findings

Established a basis extension theorem for modules over $\,\mathbb{Z}_{p^r}$

Demonstrated duality properties within these modules

Provided foundational algebraic structures for modules over prime power rings

Abstract

We present an algebraic structure in modules over integer rings with cardinality prime powers, which allows to define bases. With such structure, we prove a similar version for the basis extension theorem of linear algebra over fields. Moreover, we exhibit results involving the modules and their duals.