This article is part of the author’s work to investigate the divisibility in \(\beta \mathbb{N}\) in recent years. In this paper the author investigates the exploration of various aspects of divisibility of ultrafilters. By several ways the divisibility relation \(|\) on the natural numbers \(\mathbb{N}\) has been extended from the natural numbers to \(\beta\mathbb{N}\). Now \((\beta \mathbb{N},\cdot)\) is non-commutative, so naturally we will have three types of divisibility of ultrafilters as left \((|_L)\), right \((|_R)\) and middle \((|_M)\) divisibility. A stronger divisibility \(\widetilde{|}\) can also be defined on \(\beta \mathbb{N}\) and the notion multiplicative finite embeddability \(\leq_{fe}\) be introduced.
It is proved that \(|_L\subset \leq_{fe}\subset \tilde{|}\) and \(|_M\subset C\subset \tilde{|}\). In Section 3, the reader can find some relations between \(\leq_{fe}\)-minimal elements, prime ultrafilters and \(\mathbb{N}\)-free ultrafilters. In Section 4, the \(\tilde{|}\)-greatest class has been denoted by MAX. It is proved that MAX is a \(\cdot\)-ideal and \(+\)-closed in \(\beta\mathbb{N}\). In particular, the closure of additive idempotents is a subset of MAX. In Section 5, the reader can find a new version of classification of large sets. Finally, in Section 6 the reader can find some open problems.