On projectively generated proper classes of modules

Recently, two new techniques have been introduced for measuring the projectivity level of a module. Inspired and motivated by recent research on the topics, we initiate a study of the class of all projectively generated proper classes and denote it by pi(R) for the ring R. We analyze the class pi(R) in detail. Among other results, we show that if pi(R) is a set, there is a proper class projectively generated which contains only split short exact sequences. In a more specific case, this result has a converse. We give a characterization of rings over which pi(R) is linearly ordered, and we show that |pi(R)|not equal 3 over any ring. We prove that every proper class projectively generated by a non-projective indecomposable module is a maximal element of pi(R) over a right pure-semisimple ring. Under suitable operations, the poset (pi(R),subset of) turns out to be a complete lattice, and it is Boolean if R is a right pure-semisimple ring.