Baer criterion for injectivity in abelian categories

Abstract

It is well known that the Baer criterion for injectivity of [Formula: see text]-modules, for a ring [Formula: see text] with unit, is not true in a general category, even in a general abelian category. In this paper, we prove some results analogous to the Baer criterion for injectivity in abelian categories and Grothendieck categories. In particular, we generalize the known fact that [Formula: see text]-injectivity is the same as injectivity, if [Formula: see text] is a generator in a Grothendieck category. Furthermore, some Baer type theorems for general abelian categories are proved. Finally, equivalent conditions to satisfying a classical kind of Baer criterion are found in (locally presentable) abelian categories.