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A categorical concept of completion of objects

Guillaume C. L. Brümmer, Eraldo Giuli (1992)

Commentationes Mathematicae Universitatis Carolinae

We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.

A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

There is a classical result known as Baer’s Lemma that states that an R -module E is injective if it is injective for R . This means that if a map from a submodule of R , that is, from a left ideal L of R to E can always be extended to R , then a map to E from a submodule A of any R -module B can be extended to B ; in other words, E is injective. In this paper, we generalize this result to the category q ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...

A short proof of Eilenberg and Moore’s theorem

Maria Nogin (2007)

Open Mathematics

In this paper we give a short and simple proof the following theorem of S. Eilenberg and J.C. Moore: the only injective object in the category of groups is the trivial group.

Almost free splitters

Rüdiger Göbel, Saharon Shelah (1999)

Colloquium Mathematicae

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that E x t R ( G , G ) = 0 . For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger...

An addendum and corrigendum to "Almost free splitters" (Colloq. Math. 81 (1999), 193-221)

Rüdiger Göbel, Saharon Shelah (2001)

Colloquium Mathematicae

Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if E x t ¹ R ( G , G ) = 0 . In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

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