Ring extension of entire ring with conjugation; arithmetic in entire rings

TL;DR

This paper extends fundamental properties of integers to entire rings using a conjugation-based ring extension, exploring its algebraic structure and providing illustrative examples.

Contribution

It introduces a novel ring extension with conjugation for entire rings, generalizing integer arithmetic and analyzing its algebraic properties.

Findings

Arithmetic in entire principal rings parallels that of integers.

The ring extension with conjugation preserves multiplicative structure.

Several examples of such ring extensions are provided.

Abstract

Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties are derived from a $\star$-ring extension of the considered entire ring (ring extension with conjugation) equipped with a real function which is a multiplicative structure-preserving map between two algebras. The algebra of this ring extension is studied in detail. Some examples of such ring extension are given.