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Infinite Abelian Groups

Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

Topology and Geometry

We say a ring R is π-Baer if the right annihilator of every projection invariant left ideal of R is generated by an idempotent element of R. In this paper, we study connections between the π-Baer condition and related conditions such as the Baer, quasi-Baer and π-extending conditions. The 2-by-2 generalized triangular and the n-by-n triangular π-Baer matrix rings are characterized. Also, we prove that a n-by-n full matrix ring over a π-Baer ring is a π-Baer ring. In contrast to the Baer condition, it is shown that the π-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

π-Baer rings

Let A be a simple $$C^*$$-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of A is realized as the rank of an operator in the stabilization of A. Assuming moreover that A has locally finite nuclear dimension, we deduce that A is $$\mathcal {Z}$$-stable if and only if it has strict comparison of positive elements. In particular, the Toms–Winter conjecture holds for simple, approximately subhomogeneous $$C^*$$-algebras with stable rank one.

Ranks of Operators in Simple C ∗ -algebras with Stable Rank One

In this article, we define a module M to be &#x1d4a2;-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for &#x1d4a2;-extending modules and give a characterization of the Abelian groups which are &#x1d4a2;-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are &#x1d4a2;-extending. We prove that a minimal cogenerator for &#x1d4a2;od-R is &#x1d4a2;-extending, but not, in general, extending. It is also shown that if M is (&#x1d4a2;-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

Goldie Extending Modules

Module Theory, Extending Modules and Generalizations

A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module &#x1d49e;-extending if every member of the set &#x1d49e; is essential in a direct summand where &#x1d49e; is a subset of the set of all submodules of M. Our focus is the behavior of the &#x1d49e;-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the &#x1d49e;-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets &#x1d49e;, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups.

The Extending Condition Relative to Sets of Submodules

A module M is said to satisfy the C 11 condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C 1 condition implies the C 11 condition and that the class of C 11-modules is closed under direct sums but not under direct summands. We show that if M = M 1 ⊕ M 2, where M has C 11 and M 1 is a fully invariant submodule of M, then both M 1 and M 2 are C 11-modules. Moreover, the C 11 condition is shown to be closed under formation of the ring of column finite matrices of size Γ, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C 11 are essential extensions of C 11-modules constructed from M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C 11-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

Adnan Tercan

Papers

π-Baer rings

Goldie Extending Modules

Module Theory, Extending Modules and Generalizations

The Extending Condition Relative to Sets of Submodules

When Some Complement of a Submodule Is a Summand