On locally pure-injective modules

The subject of this article are the modules M over a ring R such that every element of M is contained in a pure-injective direct summand of M. For obvious reasons we call these modules locally pure-injective. We prove diverse characterizations, some structural results and give conditions under which locally pure-injectives are pure-injective. Furthermore, we show that the sets of matrix subgroups of the modules in question satisfy the AB5* condition. One of our characterizations reveals that the class of locally pure-injective modules is in a certain sense the dual of the class of strict Mittag–Leffler modules (Raynaud and Gruson, Invent. Math. 13 (1971) 1).