  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.
