Recent zbMATH articles in MSC 13Ahttps://zbmath.org/atom/cc/13A2023-11-13T18:48:18.785376ZUnknown authorWerkzeugMini-workshop: Subvarieties in projective spaces and their projections. Abstracts from the mini-workshop held November 27 -- December 3, 2022https://zbmath.org/1521.000092023-11-13T18:48:18.785376ZSummary: The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study.Genus two nilpotent graphs of finite commutative ringshttps://zbmath.org/1521.052172023-11-13T18:48:18.785376Z"Kalaimurugan, G."https://zbmath.org/authors/?q=ai:kalaimurugan.gnanappirakasam"Vignesh, P."https://zbmath.org/authors/?q=ai:vignesh.p"Tamizh Chelvam, T."https://zbmath.org/authors/?q=ai:tamizh-chelvam.thirugnanamConsider a finite commutative ring \(R\) with identity and denote by
\begin{itemize}
\item \(Z(R)\) the subset of the zero divisors of \(R\);
\item \(Z_N(R):=\{y \in R: \exists x \in R\setminus\{0\} \textrm{ s.t. } xy \textrm{ nilpotent in } R\}\);
\item \(I \subseteq R\) an ideal of \(R\) and
\item \(\mathrm{ann}_R(I)=\{x \in R: xI=\{0\}\}\) the annihilator ideal.
\end{itemize}
We say that an ideal \(J\) of \(R\) is essential if \(J \cap I \neq \{0\}\), for each nonzero ideal \(I\) of \(R\).
We can associate graphs to these structures:
\begin{itemize}
\item \(\Gamma(R)\): zerodivisor graph.
It is a simple and undirected graph; its vertices are the elements of \(V=Z(R) \setminus \{0\}\) and if \(x,y \in V\), there's an edge between them iff \(xy=0\).
\item \(EG(R)\): essential graph.
It is a simple and undirected graph; its vertices are the elements of \(V=Z(R) \setminus \{0\}\) and if \(x,y \in V\), there's an edge between them iff \(\mathrm{ann}_R(xy)\) is an essential ideal for \(R\).
\item \(\Gamma_N(R)\): nilpotent graph [\textit{A.-H. Li} and \textit{Q.-S. Li}, Int. J. Algebra 4, No. 5--8, 291--302 (2010; Zbl 1210.16010)].
It is a simple and undirected graph; its vertices are the elements of \(V_N=Z_N(R) \setminus \{0\}\) and if \(x,y \in V_N\), there's an edge between them iff \(xy\) is nilpotent.
\end{itemize}
Relying on some relations of [\textit{M. J. Nikmehr} et al., J. Algebra Appl. 16, No. 7, Article ID 1750132, 14 p. (2017; Zbl 1367.13005)], the paper recalls that:
\begin{itemize}
\item if \(R\) is reduced, \(\Gamma(R) \simeq EG(R) \simeq \Gamma_N(R)\);
\item if \(R\) is nonreduced, \(\Gamma(R) \subseteq EG(R) \subset \Gamma_N(R)\).
\end{itemize}
We call genus of a graph \(G\) the minimum positive integer \(g(G):=g\) such that we can embed the graph in the surface \(S_g\) without edge crossings.
The paper characterizes all finite commutative rings with identity such that their nilpotent graphs have genus two as follows
Theorem. Let \(R\) be a finite commutative ring with identity. It holds \(g(\Gamma_N(R))=2\) if and only if \(R\) is isomorphic to one of the following rings:
\begin{itemize}
\item \(\mathbb{F}_4 \times \mathbb{F}_8\)
\item \(\mathbb{F}_4 \times\mathbb{F}_9\)
\item \(\mathbb{F}_4 \times \mathbb{Z}_{11}\)
\item \(\mathbb{Z}_5 \times\mathbb{Z}_7 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathbb{F}_8\)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathbb{F}_9 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_{11}\)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_3 \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_3 \times\mathbb{Z}_3 \times \mathbb{F}_4\)
\item \(\mathbb{Z}_4 \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_2[x]/(x^2) \times\mathbb{Z}_5 \)
\item \(\mathbb{Z}_4 \times\mathbb{Z}_2 \times\mathbb{Z}_2\)
\item \(\mathbb{Z}_2[x]/(x^2) \times\mathbb{Z}_2 \times\mathbb{Z}_2 \)
\item \(\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_3\).
\end{itemize}
Reviewer: Michela Ceria (Bari)Polynomials, \( \alpha \)-ideals, and the principal latticehttps://zbmath.org/1521.060072023-11-13T18:48:18.785376Z"Molkhasi, Ali"https://zbmath.org/authors/?q=ai:molkhasi.aliSummary: Let \(R\) be a commutative ring with an identity, \( \mathfrak{R}\) be an almost distributive lattice and \(I_\alpha(\mathfrak{R})\) be the set of all \(\alpha \)-ideals of \(\mathfrak{R}\). If \(L(R)\) is the principal lattice of \(R\), then \(R[I_\alpha(\mathfrak{R})]\) is Cohen-Macaulay. In particular, \(R[I_\alpha(\mathfrak{R})][X_1,X_2,\cdots]\) is WB-height-unmixed.A characterisation of atomicityhttps://zbmath.org/1521.130022023-11-13T18:48:18.785376Z"Tringali, Salvatore"https://zbmath.org/authors/?q=ai:tringali.salvatoreThe author studies factorability in preordered monoids (not necessarily commutative or cancellative). In particular, the author proves that an acyclic monoid is atomic if and only if it has a generating set whose elements satisfy the ACCP. Hence, an integral domain is atomic if and only the multiplicative monoid of its nonzero elements is generated by a set of elements satisfying the ascending chain condition on principal ideals. This characterization answers a question of Geroldinger and Koch.
Reviewer: Moshe Roitman (Haifa)On \(2r\)-ideals in commutative rings with zero-divisorshttps://zbmath.org/1521.130032023-11-13T18:48:18.785376Z"Alhazmy, Khaled"https://zbmath.org/authors/?q=ai:alhazmy.khaled"Almahdi, Fuad Ali Ahmed"https://zbmath.org/authors/?q=ai:almahdi.fuad-ali-ahmed"Bouba, El Mehdi"https://zbmath.org/authors/?q=ai:bouba.el-mehdi"Tamekkante, Mohammed"https://zbmath.org/authors/?q=ai:tamekkante.mohammedLet \(R\) be a commutative ring with identity and \(Z(R)\) its set of zero-divizors. A proper ideal \(I\) of \(R\) is said to be a uniformly \(pr\)-ideal if there exists a positive integer \(n\) such that, whenever \(x,y\in R\) with \(xy\in I\), then \(x^n\in I\) or \(y\in Z(R)\). The order of \(I\) is the smallest positive integer for which the aforementioned property holds. The goal of this paper is to study the uniformly \(pr\)-ideals with order \(\leq 2\), which are called \(2r\)-ideals. After giving several properties and characterizations of such ideals, the authors show that many known classes of ideals are \(2r\)-ideals. They also include the study of \(2r\)-ideals in polynomials rings.
Reviewer: Ali Benhissi (Monastir)\(S\)-principal ideal multiplication moduleshttps://zbmath.org/1521.130042023-11-13T18:48:18.785376Z"Aslankarayiğit Uğurlu, Emel"https://zbmath.org/authors/?q=ai:ugurlu.emel-aslankarayigit"Koç, Suat"https://zbmath.org/authors/?q=ai:koc.suat"Tekir, Ünsal"https://zbmath.org/authors/?q=ai:tekir.unsalSummary: In this paper, we study \(S\)-Principal ideal multiplication modules. Let \(A\) be a commutative ring with \(1 \neq 0\), \(S \subseteq A\) a multiplicatively closed set and \(M\) an \(A\)-module. A submodule \(N\) of \(M\) is said to be an \(S\)-\textit{multiple} of \(M\) if there exist \(s \in S\) and a principal ideal \(I\) of \(A\) such that \(sN \subseteq IM \subseteq N\). \(M\) is said to be an \(S\)-\textit{principal ideal multiplication module} if every submodule \(N\) of \(M\) is an \(S\)-multiple of \(M\). Various examples and properties of \(S\)-principal ideal multiplication modules are given. We investigate the conditions under which the trivial extension \(A\ltimes M\) is an \(S \ltimes 0\)-principal ideal ring. Also, we prove Cohen type theorem for \(S\)-principal ideal multiplication modules in terms of \(S\)-prime submodules.Companion varieties for Hesse, Hesse union dual Hesse arrangementshttps://zbmath.org/1521.130052023-11-13T18:48:18.785376Z"De Poi, Pietro"https://zbmath.org/authors/?q=ai:de-poi.pietro"Ilardi, Giovanna"https://zbmath.org/authors/?q=ai:ilardi.giovannaIn the paper under review, the authors study the so-called companion varieties for some line arrangements over the complex numbers. The starting point is the notion of unexpected hypersurfaces.
Definition. We say that a reduced set of points \(Z \subset \mathbb{P}^{N}\) admits an unexpected hypersurface of degree \(d\) if there exists a sequence of non-negative integers \(m_{1}, \dots m_{s}\) such that for all general points \(P_{1}, \dots P_{s}\) the zero dimensional subscheme \(P=m_{1}P_{1} + \dots + m_{s}P_{s}\) fails to impose independent conditions on forms of degree \(d\) vanishing along \(Z\) and the set of such forms is non-empty.
Assume now that there is a set of points \(Z \subset \mathbb{P}^{N}\) which admits a unique unexpected hypersurface \(H_{Z,P}\) of degree \(d\) and multiplicity \(m\) at a general point \(P=(a_{0}: \dots : a_{N}) \in \mathbb{P}^{N}\). Let \[F_{Z}((x_{0}:\, \dots \,: x_{N}),(a_{0}:\, \dots \,: a_{N})) = 0\] be a homogeneous polynomial equation of \(H_{Z,P}\). Let \(g_{0}, \dots , g_{M}\) be a basis of the vector space \([I(Z)]_{d}\) of homogeneous polynomials of degree \(d\) vanishing at all points of \(Z\). Under some reasonable conditions the unexpected hypersurface \(H_{Z,P}\) comes from a bi-homogeneous polynomial \(F_{Z}((x_{0}: \dots : x_{N}),(a_{0}: \dots : a_{N}))\) of bi-degree \((m,d)\). Indeed, \(F_{Z}\) can be written in a unique way as a combination \[(\star): \quad F_{Z} = h_{0}(a_{0} : \dots : a_{N})g_{0}(x_{0} : \dots : x_{N}) + \cdots + h_{M}(a_{0} : \dots : a_{N})g_{M}(x_{0} : \dots : x_{N}),\] where \(g_{0}(x_{1}: \dots : x_{N})\), \dots , \(g_{M}(x_{0} : \dots x_{N})\) are homogeneous polynomials of degree \(d\) and \(h_{0}(a_{0} : \dots : a_{N})\), \dots , \(h_{M}(a_{0}: \dots : a_{N})\) are homogeneous polynomials of degree \(m\). Therefore, there are two rational maps naturally associated with \((\star)\), namely \[\phi : \mathbb{P}^{N} \ni (x_{0} : \dots : x_{N}) \mapsto(g_{0}(x_{0}: \dots : x_{N}): \dots : g_{M}(x_{0}: \dots : x_{N})) \in \mathbb{P}^{N},\] \[\psi : \mathbb{P}^{N} \ni (a_{0} : \dots : a_{N}) \mapsto(h_{0}(a_{0}: \dots : a_{N}): \dots : h_{M}(a_{0}: \dots : a_{N})) \in \mathbb{P}^{N}.\] The images of these maps are the companion varieties. The main result of the paper under review can be formulated as follows.
Main Theorem. The image \(S\) of \(\phi\) is a smooth arithmetically Cohen-Macaulay rational surface in the case of the Hesse and the merger of the Hesse and the dual Hesse arrangements.
1) In the case of the Hesse arrangement, the surface \(S\) is of degree \(13\). More precisely, it is the plane blow up in the \(12\) points of \(Z(\mathrm{Hesse})\) (see system (2) therein for details) embedded into \(\mathbb{P}^{8}\) with the complete linear system of the quintics through \(Z(\mathrm{Hesse})\). Its ideal \(I(S)\) is generated by \(15\) quadrics.
2) In the case of the merger of the Hesse and the dual Hesse, the surface \(S\) is of degree \(43\). More precisely, it is the plane blown-up in the \(21\) points of \(Z(\mathrm{Hesse} \cup \mathrm{dualHesse})\) (see system (6) therein for details), embedded into \(\mathbb{P}^{23}\) with the complete linear system of the octics through \(Z(\mathrm{Hesse} \cup\mathrm{dualHesse})\). Its ideal \(I(S)\) is generated by \(210\) quadrics.
Reviewer: Piotr Pokora (Kraków)FMR-rings in some distinguished constructionshttps://zbmath.org/1521.130062023-11-13T18:48:18.785376Z"Ouzzaouit, Omar"https://zbmath.org/authors/?q=ai:ouzzaouit.omar"Tamoussit, Ali"https://zbmath.org/authors/?q=ai:tamoussit.aliLet \(R\) be a commutative ring with unit. Then \(R\) is called an FMR-ring if for each maximal ideal \(M\) of \(R\), the field \(R/M\) is finite. This paper is devoted to the transfer of the FMR-ring property to the trivial ring extension and to the bi-amalgamated algebra. The main results can be summarized as follows;
The case of the trivial ring extension: Let \(R\) be a commutative ring and \(E\) be an \(R\)-module. Then \(R\propto E\) is an FMR-ring if and only if so is \(R\).
The case of the bi-amalgamated algebra: Let \(f:A\to B\) and \(g:A\to C\) be two ring homomorphisms, and let \(J\) and \(J'\) be two ideals of \(B\) and \(C\), respectively, such that \(f^{-1}(J)=g^{-1}(J')\). Then \(A\bowtie^{(f,g)}(J,J')\) is an FMR-ring if and only if so are \(f(A)+J\) and \(g(A)+J'\). If, in addition, \(f\) and \(g\) are surjective, then \(A\bowtie^{(f,g)}(J,J')\) is an FMR-ring if and only if so are \(B\) and \(C\).
Reviewer: Mohamed Aqalmoun (Fès)Essentially finite generation of valuation rings in terms of classical invariantshttps://zbmath.org/1521.130072023-11-13T18:48:18.785376Z"Cutkosky, Steven Dale"https://zbmath.org/authors/?q=ai:cutkosky.steven-dale"Novacoski, Josnei"https://zbmath.org/authors/?q=ai:novacoski.josneiSummary: The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field \((K,\nu)\) and an extension \(\omega\) of \(\nu\) to a finite extension \(L\) of \(K\). Then we study when the valuation ring of \(\omega\) is essentially finitely generated over the valuation ring of \(\nu \). We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).When does a perturbation of the equations preserve the normal cone?https://zbmath.org/1521.130082023-11-13T18:48:18.785376Z"Quy, Pham Hung"https://zbmath.org/authors/?q=ai:pham-hung-quy."Trung, Ngo Viet"https://zbmath.org/authors/?q=ai:ngo-viet-trung.Let \(I = (f_1,\ldots,f_r)\) denote an ideal of a local Noetherian ring \((R,\mathfrak{m})\). An ideal \(I' = (f_1',\ldots,f_m')\) of \(R\) is called a small pertubation of \(I\) whenever \(f_i \cong f_i' \mod \mathfrak{m}^N\) for \(i=1,\ldots,m\) and \(N \gg 0\). It is of some interest to know which properties of \(R/I\) are preserved under small pertubations. For instances when defining ideal of the singularity of an analytic space is replaced by their \(N\)-jets for some \(N \gg 0\). The paper deals with the following problem: Let \(J \subset R\) denote an arbitrary ideal. For which ideal \(I\) does there exists a number \(N\) such that \(f_i \cong f_i' \mod J^N\) for \(i=1,\ldots,m\) such that \(\operatorname{gr}_J(R/I) \cong \operatorname{gr}_J(R/I')\)? (Note that \(\operatorname{Spec} (\operatorname{gr}_J(R/I) \) is the normal cone of the blow-up of \(R/I\) along \(J\).) -- In their main result the authors provide an affirmative answer when \(f_1,\ldots,f_m\) is a \(J\)-filter regular sequence, i.e. \((f_1,\ldots,f_{i-1}) :_R f_i /(f_1,\ldots,f_{i-1})\) is of \(J\)-torsion for \(i = 1,\ldots,m\). Moreover, they prove a converse to the result by assuming that \(\operatorname{gr}_J(R/(f_1,\ldots,f_i)) \cong \operatorname{gr}_J(R/(f_1',\ldots,f_i'))\) for \(i = 1,\ldots,m\). Finally they generalize some of their results to Noetherian filtrations.
Reviewer: Peter Schenzel (Halle)Enumeration of \(\mathcal{D}\)-invariant ideals of the ring \(R_n(K,J)\)https://zbmath.org/1521.130092023-11-13T18:48:18.785376Z"Davletshin, Maksim N."https://zbmath.org/authors/?q=ai:davletshin.maksim-nikolaevichSummary: Let \(K\) be a local ring of the main ideal with a nilpotent maximal ideal \(J\). The paper is devoted to finished of solution of problem enumeration of ideals of the ring \(K\) of \(n\times n\) matrices with coefficients of \(J\) on the main diagonal and above it.Almost strongly unital ringshttps://zbmath.org/1521.130102023-11-13T18:48:18.785376Z"Oman, Greg"https://zbmath.org/authors/?q=ai:oman.greg-g"Senkoff, Evan"https://zbmath.org/authors/?q=ai:senkoff.evanAs the authors mentioned, if \(P\) is a certain property, then a mathematical structure \(\mathcal{A}\) almost has property \(P\) if \(\mathcal{A}\) does not have property \(P\), but every substructure (or quotient structure) of \(\mathcal{A}\) has property \(P\). If \(R\) is a ring (not necessarily commutative or with identity), \(S\subseteq R\) is called a subring of \(R\), if \((S,+)\) is a subgroup of \((R,+)\) and \(S\) is closed under the multiplication of \(R\). If each subring \(S\) of \(R\) has an identity, say \(1_S\) (it is possible that \(1_R\neq 1_S\)), then \(R\) is called strongly unital ring. These rings completely determined in [\textit{G. Oman} and \textit{J. Stroud}, Involve 13, No. 5, 823--828 (2020; Zbl 1479.16002)]. In fact, they proved that a nontrivial ring \(R\) is strongly unital if and only if \(R\cong F_1\times\cdots\times F_n\), where each \(F_i\) is an absolutely algebraic field (i.e., field with nonzero characteristic which is algebraic over its prime subfield).
In this article, a ring \(R\) is called almost strongly unital, if \(R\) has no identity but every proper subring of \(R\) has an identity. In Lemma 6, they prove that each commutative reduced Artinian ring has an identity. In Theorem 8, they prove that if \(R\) is a nonzero ring, then every proper subring of \(R\) has an identity if and only if either \(R\) is strongly unital or \(R\cong \frac{X\mathbb{F}_p[X]}{<X^2>}\), where \(p\) is a prime number and \(\mathbb{F}_p\) is a field with exactly \(p\) elements.
Reviewer: Alborz Azarang (Ahvaz)Erratum to: ``Rings in which every ideal contained in the set of zero-divisors is a d-ideal''https://zbmath.org/1521.130122023-11-13T18:48:18.785376Z"Anebri, Adam"https://zbmath.org/authors/?q=ai:anebri.adam"Mahdou, Najib"https://zbmath.org/authors/?q=ai:mahdou.najib"Mimouni, Abdeslam"https://zbmath.org/authors/?q=ai:mimouni.abdeslamSummary: In this erratum, we correct a mistake in the proof of Proposition 2.7 in our paper [ibid. 37, No. 1, 45--56 (2022; Zbl 1483.13022)]. In fact the equivalence \((3)\Longleftarrow (4)\) ``\(R\) is a quasi-regular ring if and only if \(R\) is a reduced ring and every principal ideal contained in \(Z(R)\) is a 0-ideal'' does not hold as we only have \(Rx\subseteq O(S)\).Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular treeshttps://zbmath.org/1521.130172023-11-13T18:48:18.785376Z"Shaukat, Bakhtawar"https://zbmath.org/authors/?q=ai:shaukat.bakhtawar"Haq, Ahtsham Ul"https://zbmath.org/authors/?q=ai:haq.ahtsham-ul"Ishaq, Muhammad"https://zbmath.org/authors/?q=ai:ishaq.muhammadSummary: Let \(S\) be a polynomial algebra over a field. If \(I\) is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of \(S/I\).Classes of trivial ring extensions and amalgamations subject to pseudo-almost valuation conditionhttps://zbmath.org/1521.130282023-11-13T18:48:18.785376Z"Moutui, Moutu Abdou Salam"https://zbmath.org/authors/?q=ai:moutui.moutu-abdou-salam"Ouled Azaiez, Najib"https://zbmath.org/authors/?q=ai:ouled-azaiez.najib"Koç, Suat"https://zbmath.org/authors/?q=ai:koc.suatSummary: This paper studies the transfer of pseudo-almost valuation property (PAVR property for short) to various context of commutative ring extensions such as power series ring, trivial ring extension and amalgamation. Our work is motivated by an attempt to generate new original classes of rings satisfying this property. The obtained results are backed with new and illustrative examples arising as trivial ring extensions, amalgamations and pullback constructions.Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domainshttps://zbmath.org/1521.130292023-11-13T18:48:18.785376Z"Hiebler, Moritz"https://zbmath.org/authors/?q=ai:hiebler.moritz"Nakato, Sarah"https://zbmath.org/authors/?q=ai:nakato.sarah"Rissner, Roswitha"https://zbmath.org/authors/?q=ai:rissner.roswithaFor a commutative ring \(A\) with identity, an irreducible element \(a \in A\) is called \textit{absolutely irreducible}, if for every \(k \in \mathbb N\) the element \(a^k\) has (up to associates) only the trivial factorization \(a \cdot a \cdot \ldots \cdot a\).
Let \(R\) be a discrete valuation domain with finite residue field \(R/pR\), where \(p\) is a prime element of \(R\), and let \(K\) denote the quotient field of \(R\). The authors investigate absolutely irreducible elements of the ring
\[
\text{Int}(R) = \{ F \in K[X] \mid F(R) \subseteq R \}
\]
of all integer-valued polynomials of \(K[X]\). Every irreducible \(F \in \text{Int}(R)\) can be written in the form \(F = f/p^n\) with some primitive polynomial \(f \in R[X]\) and \(n \ge 0\).
The authors prove that \(F\) is absolutely irreducible in \(\text{Int}(R)\) if and only if the \textit{fixed divisor kernel} of \(f\) is trivial (and \(f\) is not a proper power of a polynomial of \(R[X]\)). The notions and ideas for the proof are contained in Section 4 of the paper.
Furthermore, for given \(F\) as above, the authors determine explicite values \(s \in \mathbb N\) such that \(F\) is absolutely irreducible if and only if \(F^s\) has only the trivial factorization in \(\text{Int}(R)\). Finally they argue that these values for \(s\) are best possible.
Reviewer: Günter Lettl (Graz)Amenable groups of finite cohomological dimension and the zero divisor conjecturehttps://zbmath.org/1521.201132023-11-13T18:48:18.785376Z"Degrijse, Dieter"https://zbmath.org/authors/?q=ai:degrijse.dieterThe paper deals with amenable groups which are not elementary. The paper proves that every amenable group of cohomological dimension two (over the integers) is solvable focusing on Kaplansky's zero divisor conjecture and its generalizations. The paper's main results establish connections between cohomological dimension, finiteness properties, and the solvability of amenable groups under specific conditions. The strategy of the proof is partly based on the work of P. H. Kropholler.
Reviewer: Meral Tosun (İstanbul)