Euclidean integers, Euclidean ultrafilters, and Euclidean numerosities

We introduce axiomatically the ring $\bf{Z}_\kappa$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $\kappa$-sequences of integers is well defined. In particular any ordinal might be identified with the transfiite sum of its characteristic function, preserving the so called natural operations. The ordered ring $\bf{Z}_\kappa$ may be obtained as an ultrapower of $\mathbb{Z}$ modulo suitable ultrafilters, thus constituting a \it{ring of nonstandard integers.} Most relevant is the \it{algebraic} characterization of the ordering: a Euclidean integer is \it{positive} if and only if it is \it{the transfinite sum of natural numbers.} This property requires the use of special ultrafilters called Euclidean, here introduced to ths end. The ring $\bf{Z}_\kappa$ allows to assign a ``Euclidean" size (\it{numerosity}) to ``ordinal Punktmengen", i.e. sets of tuples of ordinals, as the transfinite sum of their characteristic functions: so every set becomes equinumerous to a set of ordinals, the Cantorian defiitions of \it{order, addition and multiplication} are maintained, while the Euclidean principle ``the whole is greater than the part" (\it{a set is (strictly) larger than its proper subsets}) is fulfilled.