Injectives and simple objects

“…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].…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…

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 < €.

…”

mentioning

confidence: 99%

“…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)]. …”

mentioning

confidence: 99%

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Subsimple, injective, retract

Feigelstock

Klein

1985

Proceedings of the Edinburgh Mathematical Society

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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.

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A categorical approach to largest and smallest connectednesses and disconnectednesses

Veldsman¹

1984

Acta Math Hung

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w IntroductionMotivated by the existence of largest and smallest proper connectednesses and disconnectednesses in the categories of topological spaces (Top) and undirected graphs with loops (Graph), we look for necessary and sufficient conditions for the existence of such classes in a general categorical setting. This leads us to consider certain classes of simple objects.We start by giving the necessary theory of radical classes (connectednesses) and semisimple classes (disconnectednesses). In the next section we define certain subclasses of the class of all simple objects. These classes are then used in the third section to obtain largest and smallest radical and semisimple classes. Lastly we generalize the results of Wiegandt [8] concerning the hereditariness and cohereditariness of connectednesses and disconnectednesses.K will always be an arbitrary category and E and M will denote subclasses of the class of K-epimorphisms and K-monomorphisms respectively. Throughout this paper we shall assume that both E and M are closed under composition with isomorphisms and that the class of all K-identities is contained in EflM. Note that these two assumptions imply that the class of all K-isomorphisms is contained in E flM. A~BCE will mean there is a morphism ~: A-~B in Kwith c~EE. A~BEM will have a similar meaning. Let T be the class of all trivial objects in K (cf.

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Injectives and simple objects

Cited by 7 publications

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On a Paper of G. Mason

On a Paper of G. Mason

Subsimple, injective, retract

A categorical approach to largest and smallest connectednesses and disconnectednesses

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