Subsimple, injective, retract

Abstract

Simple and subsimple objects were introduced in [6]. It was shown that if there are enough simple objects in a category , then there is no room for injectives in . This idea was exploited in [6] and [2] to show that several classes of groups, rings and classes belonging to other categories do not possess non-trivial injectives or retracts. In this note, the above results will be strengthened by introducing a weaker condition than subsimple of [6]. As a consequence, and by employing some embedding theorems, we show that some important classes do not possess non-trivial retracts.All the categories are assumed to have a zero object.