arXiv subjects
Publications and source records attributed to Simion Breaz.
We prove that all pure submodules of a finite rank torsion-free module $A$ over a Dedekind domain are $A$-generated if and only if $A$ has a rank $1$ direct summand $B$ such that $\mathbf{type}(B)$ is the inner type of $A$.
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcategories) in compactly generated categories and cohomological functors into Grothendieck categories.
We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular every torsion enveloping class in $\textrm{Mod-} R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.
We study the class of rings $R$ with the property that for $x\in R$ at least one of the elements $x$ and $1+x$ are tripotent.
We study the class of modules, called cosilting modules, which are defined as the categorical duals of silting module. Several characterizations of these modules and connections with silting modules are presented. We prove that Bazzoni theorem about the pure-injectivity of cotilting modules is also valid for cosilting modules.
{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it 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 {\it weakly nil-clean}.} It is also completely determined when a matrix ring over a division ring is weakly nil-clean.
We will study commuting properties of the defect functor $\mathrm{Dev}_β=\mathrm{Coker}\mathrm{Hom}_\mathcal{C}(β,-)$ associate to a homomorphism $β$ in a finitely presented category. As an application, we characterize objects $M$ such that $\mathrm{Ext}^1_\mathcal{C}(M,-)$ commutes with direct unions (i.e. direct limits of monomorphisms), assuming that $\mathcal{C}$ has a generator which is a direct sum of finitely presented projective objects.
We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $G\cong H$. Conversely, if $R$ is a Dedekind domain such that two $R$-modules $G$ and $H$ are isomorphic whenever the rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $R$ is a PID.
We prove that a right $R$-module $M$ is $Σ$-pure injective if and only if $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$. Consequently, if $R$ is a unital ring, the homotopy category $\mathbf{K}({\mathrm{Mod}\text{-} R})$ 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.
We classify by numerical invariants the finite subgroups $H$ of a primary abelian group $G$ for which every homomorphism or monomorphism of $H$ into $G$, or every endomorphism of $H$, extends to an endomorphism of $G$. We apply these results to show that for finitely generated subgroups of general abelian groups, the extendibility of monomorphisms implies the extendibility of all homomorphisms.
We will use Watts's theorem together with Lenzing's characterization of finitely presented modules via commuting properties of the induced tensor functor in order to study commuting properties of Ext-covariant functors.
We prove in ZFC that if $G$ is a (right) $R$-module such that the groups $\Hom_R(\prod_{i\in I}G_i,G)$ and $\prod_{i\in I}\Hom_R(G_i,G)$ are naturally isomorphic for all families of $R$-modules $(G_i)_{i\in I}$ then G=0. The result is valid even we restrict to families such that $G_i\cong G$ for all $i\in I$.