  We continue the research of an extension $\widetilde{\mid}$ of the
divisibility relation to the Stone-\v Cech compactification $\beta N$. First we
prove that ultrafilters we call prime actually possess the algebraic property
of primality. Several questions concerning the connection between
divisibilities in $\beta N$ and nonstandard extensions of $N$ are answered,
providing a few more equivalent conditions for divisibility in $\beta N$.
Results on uncountable chains in $(\beta N,\widetilde{\mid})$ are proved and
used in a construction of a well-ordered chain of maximal cardinality. Finally,
we consider ultrafilters without divisors in $N$ and among them find the
maximal class.
