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In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory. [ABSTRACT FROM AUTHOR]

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients. [ABSTRACT FROM AUTHOR]

Given a hereditary torsion theory τ = (T,F) in Mod-R, a module M is called τ-supplemented if every submodule A of M contains a direct summand C of M with A/C τ-torsion. A submodule V of M is called τ - supplement of U in M if U + V = M and U ∩ V ≤(V) and M is τ - weakly supplemented if every submodule of M has a τ-supplement in M. Let M be a τ-weakly supplemented module. Then M has a decomposition M = M1 ⊕ M2 where M1 is a semisimple module and M2 is a module with τ(M2) ≤e M2. Also, it is shown that; any finite sum of τ-weakly supplemented modules is a τ-weakly supplemented module. [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]

This article introduces the concept of a τ-supplemented module as follows: Given a hereditary torsion theory in Mod R with associated torsion functor τ, we say that a module M is τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. We present here some fundamental properties of this class of modules and study the decompositions of τ-supplemented modules under certain conditions on modules. The question of which direct sum of τ-supplemented R-modules are τ-supplemented is treated here. [ABSTRACT FROM AUTHOR]