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This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors. The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic. [ABSTRACT FROM AUTHOR]

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]

In this paper we introduce and investigate the latticial counterpart of the module-theoretical concept of preradical. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories. [ABSTRACT FROM AUTHOR]

Abstract: Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring. [Copyright &y& Elsevier]

Abstract: A Grothendieck category is said to be locally finitely generated if the subobject lattice of every object in is compactly generated, or equivalently, if possesses a family of finitely generated generators. Every nonzero locally finitely generated Grothendieck category possesses simple objects. We shall call a Grothendieck category indecomposable if is not equivalent to a product of nonzero Grothendieck categories . In this paper an example of an indecomposable nonlocally finitely generated Grothendieck category possessing simple objects is constructed, answering in the negative a sharper form of a question posed by Albu, Iosif, and Teply in [T. Albu, M. Iosif, M.L. Teply, Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005) 52–79]. [Copyright &y& Elsevier]

Abstract: Let A be a pseudocompact (or profinite) algebra, so where C is a coalgebra. We show that the if the semiartinian part (the “Dickson” part) of every A-module M splits off in M, then A is semiartinian, giving thus a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith. [Copyright &y& Elsevier]

Abstract: In [R. Colpi, K.R. Fuller, Tilting objects in abelian categories and quasitilted rings, Trans. Amer. Math. Soc., in press] tilting objects in an arbitrary abelian category are introduced and are shown to yield a version of the classical tilting theorem between and the category of modules over their endomorphism rings. Moreover, it is shown that given any faithful torsion theory in , for a ring R, the corresponding Heart is an abelian category admitting a tilting object which yields a tilting theorem between the Heart and . In this paper we first prove that is a prototype for any abelian category admitting a tilting object which tilts to in . Then we study AB-type properties of the Heart and commutations with direct limits. This allows us to show, for instance, that any abelian category with a tilting object is AB4, and to find necessary and sufficient conditions which guarantee that is a Grothendieck or even a module category. As particular situations, we examine two main cases: when is hereditary cotilting, proving that is Grothendieck and when is tilting, proving that is a module category. [Copyright &y& Elsevier]

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]