Primitive normality and primitive connectedness of the class of injective $S$-acts

The paper deals monoids over which the class of all injective $S$-acts is primitive normal and primitive connected. The following results are proved: the class of all injective acts over any monoid is primitive normal; the class of all injective acts over a right reversible monoid $S$ is primitive connected iff $S$ is a group; if a monoid $S$ is not a group and the class of all injective acts is primitive connected, then a maximal (w.r.t. inclusion) proper subact of ${}_SS$ is not finitely generated.