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A semi-localization of a category is a full reflective subcategory with the property that the reflector is semi-left-exact. There are many interesting examples of semi-localizations, as for instance any torsion-free subcategory of a semi-abelian category. By specializing a result due to S. Mantovani, we first characterize the categories which are semi-localizations of exact Mal'tsev categories. We then prove a new characterization of protomodular categories in terms of binary relations, allowing us to obtain an abstract characterization of the semi-localizations of exact protomodular categories. This result is very useful to study the (hereditarily)-torsion-free subcategories of semi-abelian categories. Some examples are considered in detail in the categories of groups, crossed modules, commutative rings and topological groups. We finally explain how these results extend the corresponding ones obtained in the abelian context by W. Rump. [ABSTRACT FROM AUTHOR]

Abstract: Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class . First, we show that a finitely generated module over a ring from the class splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from and prove that this dimension has all the nice properties of the dimension studied in [W. Lück, J. Reine Angew. Math. 495 (1998) 135–162] for finite von Neumann algebras. This dimension defines a torsion theory that we prove to be equal to the Goldie and Lambek torsion theories. Moreover, every finitely generated module splits in this torsion theory. If R is a ring in , we can embed it in a canonical way into a regular ring Q also in . We show that is isomorphic to by producing an explicit isomorphism and its inverse of monoids that extends to the isomorphism of and . [Copyright &y& Elsevier]