  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.
