Localizations of torsion-free Abelian groups. II.
A homomorphism \(\alpha\colon A\to B\) between Abelian groups \(A,B\) is called a localization of \(A\) if every homomorphism \(\varphi\) from \(A\) to \(B\) has a unique extension to an endomorphism \(\psi\) of \(B\) in the sense that \(\varphi=\psi\circ\alpha\). More generally, the author calls an Abelian group \(C\) perpendicular to \(\alpha\) if for every homomorphism \(\varphi\) from \(A\) to \(C\) there is a unique \(\psi\) from \(B\) to \(C\) such that \(\varphi=\psi\circ\alpha\). Hence \(B\) is perpendicular to \(\alpha\) if and only if \(\alpha\) is a localization of \(A\) and in this case \(B\) is called a localization of \(A\). Each localization gives rise to an idempotent localization functor \(L_\alpha\) from the category of Abelian groups into the reflective subcategory of all Abelian groups perpendicular to \(\alpha\). Furthermore, there is a canonical homomorphism \(\eta_Y\colon Y\to L_\alpha(Y)\) such that \(\eta_Y\) is perpendicular to \(X\) for all \(X\) which are perpendicular to \(\alpha\) and \(\eta_A=\alpha\). This homomorphism \(\eta_Y\) is called the coaugmentation map of \(Y\). Localizations arose in general category theory and have received a lot of attention recently, especially in the category of Abelian groups. For instance it was shown that localizations of the group of integers are exactly the so-called \(E\)-rings which had been introduced by Schultz. In this paper the author studies localizations \(\alpha\colon L\to M\) where \(L\) is a subgroup of the additive group of rational numbers \(\mathbb{Q}\) and \(M\) is torsion-free. It is proved that there are drastic differences depending on whether or not \(L\) is a subring of \(\mathbb{Q}\). If \(L\) is a ring, then \(M\) is simply an \(E\)-ring which is also an \(L\)-module. In this case the author describes the localization functor explicitly. If \(L\) is not a subring but \(M=M(\tau)\) where \(\tau\) is the type of \(L\), then there exists an \(E\)-ring \(H\) which is an \(\text{End}(L)\)-module such that \(M=L\otimes H\). However, the author provides an example of a finite rank Butler group which is a localization of \(L\) such that \(M\not=M(\tau)\) if \(L\) is not a subring. In this case it remains open to describe the groups that are perpendicular to \(\alpha\). Finally, the author gives an example of an \(E\)-ring that has a localization which is not an \(E\)-ring. This paper together with its predecessor [J. Algebra 278, No. 1, 411-429 (2004; Zbl 1067.20070)] contains nice results. Moreover, it provides an easy accessible introduction to the theory of localizations in the category of Abelian groups. Anyone who is interested in localizations should have the paper in her/his files.
scientific article; zbMATH DE number 2153327