More number theory in $\beta N$

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.