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.Definition, Let <€ be a full subcategory of a category 2). An object A of <& will be called Qi-subsimple if there exist Seob®, Teob# such that A is a proper subobject of S, S is a subobject of T, and S is simple in 3) [6, Definition (i)].Obviously a subsimple object in a category ( €, as defined in [6], is a ^-subsimple object.Theorem 1 Theorem 1. If a non-zero Ieob'tf is an extremal quotient in 2>of& @-subsimple object AeobW, then I is not injective in ( €.Proof. Assume / is injective in ( €. Let A >-* S >-> T, m non-invertible, S simple in 3), Teob<#, and let A-Z*I be extremal [4, 17.9]. As / is injective in <€ and hme^{A,T), there exists fe^(T,I) such that f(hm) = e. Clearly fh is a monomorphism since fh^O, as 7^0. Hence fh is invertible, since e is extremal. So m is an extremal epimorphism and a monomorphism, hence invertible. Contradiction.