Weak precovering of acts over semigroups

Let S be a monoid and X a class of S-acts which is closed under coproducts. The object of this paper is to find conditions under which all S-acts have X-precovering, especially when X is the class of injective (C-injective) acts. We have shown that the existence of injective precovering over Noetherian monoids implies the existence of injective covering. This work is an attempt to further facilitate the study of the conjecture that all S-acts have flat cover.