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.
Categories: math.HO