Recent zbMATH articles in MSC 20Khttps://zbmath.org/atom/cc/20K2023-11-13T18:48:18.785376ZWerkzeugOn universal modules with pure embeddingshttps://zbmath.org/1521.030882023-11-13T18:48:18.785376Z"Kucera, Thomas G."https://zbmath.org/authors/?q=ai:kucera.thomas-g"Mazari-Armida, Marcos"https://zbmath.org/authors/?q=ai:mazari-armida.marcosSummary: We show that certain classes of modules have universal models with respect to pure embeddings: Let \(R\) be a ring, \(T\) a first-order theory with an infinite model extending the theory of \(R\)-modules and \(\mathbf{K}^T = (\mathrm{Mod} (T), \leq_{\mathrm{pp}})\) (where \(\leq_{pp}\) stands for ``pure submodule''). Assume \(\mathbf{K}^T\) has the joint embedding and amalgamation properties. If \(\lambda^{| T |} = \lambda\) or \(\forall \mu < \lambda (\mu^{| T |} < \lambda)\), then \(\mathbf{K}^T\) has a universal model of cardinality \(\lambda\). As a special case, we get a recent result of \textit{S. Shelah} [Notre Dame J. Formal Logic 58, No. 2, 159--177 (2017; Zbl 1417.03231), 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of \(R\)-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer [\textit{M. Mazari-Armida}, Ann. Pure Appl. Logic 171, No. 1, Article ID 102723, 17 p. (2020; Zbl 1480.03019), Question 4.25]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.
{{\copyright} 2021 Wiley-VCH GmbH}Elements of high order in finite fields specified by binomialshttps://zbmath.org/1521.110782023-11-13T18:48:18.785376Z"Bovdi, V."https://zbmath.org/authors/?q=ai:bovdi.victor-a"Diene, A."https://zbmath.org/authors/?q=ai:diene.adama"Popovych, R."https://zbmath.org/authors/?q=ai:popovych.roman-o|popovych.roman-bSummary: Let \(\mathbb F_q\) be a field with \(q\) elements, where \(q\) is a power of a prime number \(p\geq 5\). For any integer \(m\geq 2\) and \(a\in \mathbb F_q^*\) such that the polynomial \(x^m-a\) is irreducible in \(\mathbb F_q[x]\), we combine two different methods to explicitly construct elements of high order in the field \(\mathbb F_q[x]/\langle x^m-a\rangle \). Namely, we find elements with multiplicative order of at least \(5^{\sqrt[3]{m/2}}\), which is better than previously obtained bound for such family of extension fields.The Pierce decomposition and Pierce embedding of endomorphism rings of abelian \(p\)-groupshttps://zbmath.org/1521.201162023-11-13T18:48:18.785376Z"Goldsmith, Brendan"https://zbmath.org/authors/?q=ai:goldsmith.brendan"Salce, Luigi"https://zbmath.org/authors/?q=ai:salce.luigiAll groups considered in the paper under this review are assumed to be additively written abelian \(p\)-groups. The paper is based on the fundamental paper by \textit{R. S. Pierce} [``Homomorphisms of primary abelian groups'', in: Topics in abelian groups. Chicago, IL: Scott, Foresman \& Co. 215--310 (1963)]. The authors investigate the Pierce decomposition of the endomorphism ring \(\mathrm{End}(G) =\hat{F}\oplus\mathrm{End}_s(G) \) of an abelian \(p\)-group \(G\) and its application to the recent studies of groups with minimal full inertia and of thick-thin groups. They improve one theorem from [\textit{B. Goldsmith} at al., Period. Math. Hung. 85, No. 1, 1--13 (2022; Zbl 1513.20059)] and prove the following theorem:
Theorem. Let \(G\) be a separable semi-standard \(p\)-group such that in the Pierce decomposition \(\mathrm{End}(G) =A\oplus\mathrm{End}_s(G) \), \(A\) is the completion in the \(p\)-adic topology of a free \(J_p\)-module \(F\) of rank strictly smaller than the continuum. If \(H\) is a countably infinite subgroup of \(G\), then the higher socles \(H^F[p^k]\) (\(k \geq1\)) of the subgroup \(H^F = \sum_{\alpha\in F}\alpha(H)\) are fully inert in \(G\), but not commensurable with any fully invariant subgroup of \(G\).
It is investigated if the Pierce embedding \(\Psi :\mathrm{End}(G)/H(G)\rightarrow\prod\limits_n M_{f_n(G)}\) is surjective and if it is not surjective.
Reviewer: Nikolay I. Kryuchkov (Ryazan)Multiplication groups of abelian torsion-free groups of finite rankhttps://zbmath.org/1521.201172023-11-13T18:48:18.785376Z"Kompantseva, E. I."https://zbmath.org/authors/?q=ai:kompantseva.ekaterina-igorevna"Tuganbaev, A. A."https://zbmath.org/authors/?q=ai:tuganbaev.askar-aLet \(\mathcal A_0\) be the class of torsion-free reduced block-rigid almost completely decomposable groups of ring type with cyclic regulator quotient. \(\mathcal A_0\) is one of the few classes of torsion-free abelian groups with a useful classification, see [\textit{A. Mader}, Almost completely decomposable groups. Amsterdam: Gordon and Breach (2000; Zbl 0945.20031)].
In this paper, the authors study the multiplication group Mult\((G)=G\otimes G\) of a group \(G\in \mathcal A_0\) and the consequent properties of rings on \(G\). Their principal result is that Mult\((G)\) is also in \(\mathcal A_0\). They derive its isomorphism and quasi-isomorphism invariants such as rank, regulator, regulator index, main decomposition and standard representation in terms of the corresponding invariants of \(G\).
Reviewer: Phillip Schultz (Perth)Forcing a basis into \(\aleph_1\)-free groupshttps://zbmath.org/1521.201182023-11-13T18:48:18.785376Z"Bossaller, Daniel"https://zbmath.org/authors/?q=ai:bossaller.daniel-p"Herden, Daniel"https://zbmath.org/authors/?q=ai:herden.daniel"Pasi, Alexandra V."https://zbmath.org/authors/?q=ai:pasi.alexandra-vSummary: In this paper, we address the question of when a non-free \(\aleph_1\)-free group \(H\) can be free in a transitive cardinality-preserving model extension. Using the \(\Gamma\)-invariant, denoted \(\Gamma(H)\), we present a necessary and sufficient condition resolving this question for \(\aleph_1\)-free groups of cardinality \(\aleph_1\). Specifically, if \(\Gamma(H) = [\aleph_1]\), then \(H\) will be free in a transitive model extension if and only if \(\aleph_1\) collapses, while for \(\Gamma(H) \neq [\aleph_1]\) there exist cardinality-preserving forcings that will add a basis to \(H\). In particular, for \(\Gamma(H) \neq [\aleph_1]\), we provide a poset \((\mathcal{P}_{\mathrm{pb}}, \leqslant)\) of partial bases for adding a basis to \(H\) without collapsing \(\aleph_1\).Projective modules over the ring of pseudorational numbershttps://zbmath.org/1521.201192023-11-13T18:48:18.785376Z"Timoshenko, Egor A."https://zbmath.org/authors/?q=ai:timoshenko.egor-aleksandrovichSummary: We prove the structure theorems which give a full description of projective modules over the ring of pseudorational numbers. We construct a complete system of invariants for such modules.Perfect state transfer on bi-Cayley graphs over abelian groupshttps://zbmath.org/1521.810402023-11-13T18:48:18.785376Z"Wang, Shixin"https://zbmath.org/authors/?q=ai:wang.shixin"Feng, Tao"https://zbmath.org/authors/?q=ai:feng.tao.1Summary: The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.