Abstract. Generalizing the notion of nil cleanness from [9], in parallel to [8], we define the concept of weak nil cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.
We characterize the nil clean matrix rings over fields. As a by product, we obtain a complete characterization of the finite rank Abelian groups with nil clean endomorphism ring and the Abelian groups with strongly nil clean endomorphism ring, respectively. * S. Breaz is supported by the CNCS-UEFISCDI grant PN-II-RU-TE-2011-3-0065. †
We study the class of modules, called cosilting modules, which are defined as the categorical duals of silting modules. Several characterizations of these modules and connections with silting modules are presented. We prove that Bazzoni's theorem about the pure-injectivity of cotilting modules is also valid for cosilting modules.
This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.2000 Mathematics subject classification: primary 20K21.
We prove that a right R-module M is Σ-pure injective if and only if Add(M ) ⊆ Prod(M ). Consequently, if R is a unital ring, the homotopy category K(Mod-R) satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply the main result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman.
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