Recent zbMATH articles in MSC 16Shttps://www.zbmath.org/atom/cc/16S2021-07-10T17:08:46.445117ZWerkzeugScaffolds: a graph-theoretic tool for tensor computations related to Bose-Mesner algebrashttps://www.zbmath.org/1462.053622021-07-10T17:08:46.445117Z"Martin, William J."https://www.zbmath.org/authors/?q=ai:martin.william-jSummary: We introduce a pictorial notation for certain tensors arising in the study of association schemes, based on earlier ideas of Terwilliger, Neumaier and Jaeger. These tensors, which we call ``scaffolds'', obey a simple set of rules which generalize common linear-algebraic operations such as trace, matrix product and entrywise product. We first study an elementary set of ``moves'' on scaffolds and illustrate their use in combinatorics. Next we re-visit results of Dickie, Suzuki and Terwilliger. The main new results deal with the relationships among vector spaces of scaffolds with edge weights chosen from a fixed coherent algebra and various underlying diagrams. As one consequence, we provide simple descriptions of the Terwilliger algebras of triply regular and dually triply regular association schemes. We finish with a conjecture connecting the duality of Bose-Mesner algebras to the graph-theoretic duality of circular planar graphs.Construction of free differential algebras by extending Gröbner-Shirshov baseshttps://www.zbmath.org/1462.130272021-07-10T17:08:46.445117Z"Li, Yunnan"https://www.zbmath.org/authors/?q=ai:li.yunnan"Guo, Li"https://www.zbmath.org/authors/?q=ai:guo.li|guo.li.2|guo.li.1Let \(\mathbf{k}\) be a field of characteristic zero and \(\lambda\in\mathbf{k}\). Then a differential \(\mathbf{k}\)-algebra of weight \(\lambda\) (also called a \(\lambda\)-differential \(\mathbf{k}\)-algebra) is defined as an associative \(\mathbf{k}\)-algebra\(R\) together with a linear operator \(d:R\rightarrow R\) (called a \(\lambda\)-derivation) such that \(d(xy) = d(x)y+xd(y)+\lambda d(x)d(y)\) for all \(x, y\in R\). If \(R\) is unital, it is also required that \(d(1_{R})=0\). If \(X\) is a set, then the free (\(\lambda\)-) differential algebra on \(X\) is a \(\lambda\)-differential \(\mathbf{k}\)-algebra \((\mathcal{D}_{\lambda}(X), d_{X})\) together with the map \(i_{X}:X\rightarrow \mathcal{D}_{\lambda}(X)\) satisfying the following universal property: for any \(\lambda\)-differential algebra \((R, d_{R})\) and map \(f:X\rightarrow R\), there exists a unique homomorphism of \(\lambda\)-differential algebras \(\bar{f}:D_{\lambda}(X)\rightarrow R\) such that \(f=\bar{f}\circ i_{X}\). In this paper the authors prove composition-diamond lemmas on Gröbner-Shirshov bases for associative algebras and (\(\lambda\)-) differential algebras, including their commutative counterparts. Then the composition-diamond lemma for differential algebras is applied to obtain Gröbner-Shirshov bases for free differential algebras on algebras in both the noncommutative and commutative cases. These results, in turn, allowed the authors to obtain canonical bases for the mentioned free objects. It is shown that a Gröbner-Shirshov basis of an algebra can be ``differentially'' extended to a differential Gröbner-Shirshov basis of the free differential algebra on this algebra, in all the cases except for the ``classical'' one, namely for differential commutative algebras with weight zero, when there are obstructions to such an extension. As demonstrations, several examples are given in the cases of free differential algebras on special algebras, including commutative algebras with one generator and some finite group algebras.On irreducible components of Rapoport-Zink spaceshttps://www.zbmath.org/1462.140282021-07-10T17:08:46.445117Z"Mieda, Yoichi"https://www.zbmath.org/authors/?q=ai:mieda.yoichiLet \(p>2\) be a prime number, let \(\mathcal{M}\) be a Rapoport-Zink space as defined in [\textit{M. Rapoport} and \textit{Th. Zink}, Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] and let \(\mathbb{X}\) be the \(p\)-divisible group over \(\bar{\mathbb{F}}_p\) used as ``framing object'' for the construction of \(\mathcal{M}\). The space \(\mathcal{M}\) is a formal scheme over the ring of integers of a finite extension of the completion of the maximal unramified extension of \(\mathbb{Q}_p\); it admits an action of the group \(J\) of self-quasi-isogenies of \(\mathbb{X}\) that preserve the additional structure prescribed by the datum defining \(\mathcal{M}\). The main theorem of this paper is that, assuming that the isogeny class of \(\mathbb{X}\) is associated to an abelian variety (in a sense which takes the additional structure into account), then the action of \(J\) on the set of irreducible components of the underlying reduced scheme \(\bar{\mathcal{M}}\) of \(\mathcal{M}\) has finite orbits. Note that here, in the datum defining \(\mathcal{M}\), are allowed ramified setups. In order to prove this result, the author generalises some methods of Mantovan and Oort (cf [\textit{E. Mantovan}, Astérisque 291, 201--331 (2004; Zbl 1062.11036); \textit{E. Mantovan}, Duke Math. J. 129, No. 3, 573--610 (2005; Zbl 1112.11033); \textit{F. Oort}, J. Am. Math. Soc. 17, No. 2, 267--296 (2004; Zbl 1041.14018)]) from the unramified setting to the general case. More precisely, the idea is to define suitable ``Igusa varieties'', that allow to relate \(\bar{\mathcal{M}}\) to a Newton stratum of the special fibre of an appropriate moduli space of abelian varieties with PEL structure (this explains the assumption on the isogeny class of \(\mathbb{X}\)). It is then exploited the fact that this Newton stratum is a scheme of finite type over \(\bar{\mathbb{F}}_p\).
The conclusion above finds application in studying the compactly supported \(l\)-adic cohomology \(H^i_c(M_\infty)\) of the Rapoport-Zink tower \(\{M_{K'}\}_{K'}\) associated to \(\mathcal{M}\) (a projective system of étale coverings of the generic fibre \(M\) of \(\mathcal{M}\)). This carries an action of \(J\times G'\), where \(G'\) are the \(\mathbb{Q}_p\)-valued points of a certain inner form of the reductive group \(G\) over \(\mathbb{Q}_p\) associated to \(\mathcal{M}\). Generalising Fargues' work in the unramified case [\textit{L. Fargues}, Astérisque 291, 1--199 (2004; Zbl 1196.11087)], the author deduces from the main theorem that, under the same assumption on \(\mathbb{X}\), for every compact open subgroup \(K'\) of \(G'\), the \(K'\)-invariant part \(H^i_c(M_\infty)^{K'}\) of \(H^i_c(M_\infty)\) is a finitely generated \(J\)-representation.
Regarding the assumption on which the above results depend, it can be seen as a generalisation of the classical Manin's problem. The paper provides an analysis of the quaternion unitary case, in which it is observed that the condition is always satisfied. Combining this with the above conclusions and the duality theorem from [\textit{P. Scholze} and \textit{J. Weinstein}, Berkeley lectures on \(p\)-adic geometry. Princeton, NJ: Princeton University Press (2020; Zbl 07178476)], the author deduces a more advanced result when \(\mathcal{M}\) is the basic Rapoport-Zink space for \(G\) a symplectic similitude group, in which case \(G'=G\) and \(J\) is a quaternion unitary similitude group. Namely, for all integers \(i,r\ge0\) and every irreducible smooth representation \(\rho\) of \(J\), the \(G\)-representation \(\mathrm{Ext}^r_J(H^i_c(M_\infty),\rho)^{\mathcal{D}_c-\text{sm}}\) has finite length (here, the superscript \(\mathcal{D}_c-\text{sm}\) denotes a certain operator on smooth \(G\)-representations which, if \(r=0\), coincides with taking the smooth vectors).Laurent series rings and related ringshttps://www.zbmath.org/1462.160012021-07-10T17:08:46.445117Z"Tuganbaev, Askar"https://www.zbmath.org/authors/?q=ai:tuganbaev.askar-aThe reviewed book is devoted to the intensive study of Laurent series rings over an associative (not necessarily commutative) ring with non-zero identity element. Some closely related classes of rings, such are Laurent rings and Malcev-Neumann rings, are also discussed here.
The book is divided into seventeen sections each of which contains enough interesting material to initiate a proper discussion. In order to give a more detailed and helpful information to the interested reader, we shall briefly comment on the different sections separately as follows:
Section 1. This section contains a material devoted to some preliminary properties of skew Laurent series rings and modules. Precisely, inner automorphisms, special groups and their subgroups as well as some special constructed rings are explored here.
Section 2. This section is pertained to Noetherian skew Laurent series rings. The most central is Lemma 2.6 which gives some useful criteria in this branch. The author also recalls here two critical theorems such as Theorem 2.8 and Theorem 2.9 concerning the Jacobson radical of such rings.
Section 3. This section uses the machinery of serial and Bézout rings. Concretely, reductable and non-reductable sums of submodules, modules over semi-local rings, distributive modules and rings as well as Bézout rings and principal right ideal rings are investigated, too. The section ends with some unsettled problems.
Section 4. This section contains a treatment of prime and semi-prime skew Laurent series rings. Here the author first recalls some more fundamental properties of modules and rings and then discusses what occurs in the case of Laurent series rings.
Section 5. This section treats the regular and bi-regular Laurent series rings. The author firstly recollects some backgrounds from regular and strongly regular modules and rings and then applies these fundamentals to the so-termed \(\varphi\)-reduced rings.
Section 6. This section analyzes some equivalent definitions of Laurent rings. Specifically, pseudo-differential operator rings and the consistency of generalized infinite sums with formal infinite sums in the ring \(A((x))\) are treated.
Section 7. This section considers the generalized Laurent rings and properties of generalized infinite sums. Moreover, some notation and definitions for generalized Laurent rings and properties of such rings comparing them with the characteristic properties of so-called normed rings are also considered.
Section 8. This is a short section in which the author considers some elementary properties of Laurent rings such as the coefficients and constant terms of these rings as well as the mapping \(2^A\to 2^R\), where \(R\) is a Laurent ring with coefficient ring \(A\).
Section 9. It is one of the most long sections in the book, containing enough important material on examples and relationships of Laurent rings. Various preliminaries and technicalities are given which substantiate the main results in the section. Indeed, in Subsection 9.5 skew Laurent series with skew derivation are considered, an explicit formula for multiplication of two Laurent series is stated in Subsection 9.9, rings of \(n\)-adic integers and fractional \(n\)-adic numbers are subsequently discussed in Subsection 9.10 and, finally, an example of a generalized Laurent ring which is not a Laurent ring is constructed explicitly in Subsection 9.11.
Section 10. It is one of the most interesting sections dealing with Noetherian and Artinian Laurent rings. In fact, in Subsection 10.2 is given the theorem on Noetherian Laurent rings, whereas the parallel theorem on Artinian Laurent rings is given in Subsection 10.3. Two additional remarks are also provided, namely a remark on generalized Laurent Noetherian (resp., generalized Laurent Artinian) rings as well as on generalized Laurent domains in the spirit of some already well-known results from previous sections of the book.
Section 11. It is concentrated in the study of simple and semi-simple Laurent rings. The section starts with some simple but useful lemmas and after that the author states basic results listed as Theorems 11.4, 11.5, 11.7, 11.8, 11.9, which emphasize some structural characterizations of such rings.
Section 12. It is devoted to the study of universal and serial Laurent rings. The section begins with some conventions such as definitions and remarks which both introduce the reader in this specific matter. The main results are Theorems 12.3, 12.4 and 12.6 plus Corollary 12.7.
Section 13. It concerns semi-local Laurent rings. In Theorems 13.5, 13.7 and 13.8 are distributed the most important claims. In Subsection 13.6 are given some valuable remarks, and Subsection 13.9 contains some still open questions of some interest and importance.
Section 14. It treats the connection between filtrations and (generalized) Malcev-Neumann rings. The author first considers filtered rings, ordered groups and filtrations and then compares some crucial properties of strongly filtered rings, Malcev-Neumann rings and generalized Malcev-Neumann rings, respectively. The results are presented into a few assertions (from Lemma 14.7 to Proposition 14.11).
Section 15. It proposes numerous interesting properties of generalized Malcev-Neumann rings. They are selected into a few statements (from Lemma 15.1 to Proposition 15.6).
Section 16. It suggests several interesting properties and examples of Malcev-Neumann rings. It is established here that the skew Laurent series ring is a particular case of a Malcev-Neumann series ring as well as that the latter one is just a Malcev-Neumann ring in its traditional form (see Subsection 16.9).
Section 17. It deals with Laurent series in two variables and the main result (namely, Theorem 17.6) gives a criterion when such a ring is a domain. A remark on lowest terms of Laurent series in several variable is also given here.
The book ends with a complete bibliography, a list of used notations and an index of some key words and phrases. No authors' index is given at the end.
To the reviewer's opinion, the book is very well written by Askar Tuganbaev, who is a distinguished expert in the field, and the book being an in-depth investigation of the present subject, will definitely be quite useful for too many researchers of different areas of the algebra.Hochschild cohomology for algebrashttps://www.zbmath.org/1462.160022021-07-10T17:08:46.445117Z"Witherspoon, Sarah J."https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jHochschild cohomology is the standard cohomology theory of associative algebras.
Unlike in the case of its close relatives, cohomology of groups and of Lie
algebras, so far there was no book devoted to it exclusively.
Of course, Hochschild (co)homology was treated earlier as part of books and
surveys, devoted either to different aspects of associative algebras
(perhaps, the most prominent example is \textit{R. S. Pierce}'s [Associative algebras. New York-Heidelberg-Berlin: Springer-Verlag (1982; Zbl 0497.16001)], or to homological algebra (like the classical treatises
of \textit{H. Cartan} and \textit{S. Eilenberg} [Homological algebra. Princeton, New Jersey: Princeton University Press (1956; Zbl 0075.24305)] and \textit{S. MacLane} [Homology. New York-Heidelberg-Berlin: Springer-Verlag (1963; Zbl 0133.26502)], or more
recent, in well-known books and surveys like \textit{C. A. Weibel}'s [An introduction to homological algebra. Cambridge: Cambridge University Press (1994; Zbl 0797.18001)], or
[\textit{M. Gerstenhaber} and \textit{S. D. Schack}, in: Deformation theory of algebras and structures and applications, Nato Adv. Study Inst., Castelvecchio-Pascoli/Italy 1986, Nato ASI Ser., Ser. C 247, 11--264 (1988; Zbl 0676.16022)]. Curiously enough, one of these books, namely \textit{J.-L. Loday}'s [Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)], is devoted to a more specific (co)homology
theory of associative algebras, which appeared much later than, and is based on
Hochschild cohomology.
Now, finally, this gap in the literature is filled, and we have a book devoted
exclusively to Hochschild cohomology and its applications. The author deals
mainly with cohomology of algebras over a field. Hochschild \emph{homology} is
considered only occasionally.
The book is designed to serve both as a textbook for a graduate course, and as a
reference monograph.
The first chapter starts with the classical definition of bar resolvent, and of
Hochschild cohomology and homology, then proceeds with interpretation of
cohomology in low degrees (invariants, derivations, square-zero extensions), the
cup product, the Gerstenhaber algebra structure on cohomology, and the Harrison
cohomology (the standard cohomology of commutative associative algebras).
Chapter 2 explores the cup product in greater detail and greater generality, in
particular, defines the cup product using resolutions different from the bar
resolution. The action of \(HH^*(A)\) on \(HH^*(A,B)\), where \(B\) is an
\(A\)-bimodule, is described.
In Chapter 3, Hochschild cohomology of algebras arising from some important
constructions is considered, via explicit resolutions: the tensor product of
algebras and its generalizations (the Künneth formula), monomial algebras,
smooth commutative algebras (the Hochschild-Kostant-Rosenberg theorem), Koszul
algebras, skew group algebras, and path and monomial algebras.
Chapter 4 treats Hochschild dimension (also sometimes called cohomological
dimension), algebras of cohomological dimension \(\le 1\) (called by the author
quasi-free, or Cuntz-Quillen smooth), and Calabi-Yau algebras.
Chapter 5 treats deformation theory of associative algebras à la Gerstenhaber.
The Poincaré-Birkhoff-Witt theorem is proved as an application of a more
general result about filtered deformations of Koszul algebras, due to Braverman
and Gaitsgory.
Chapter 6 describes the Gerstenhaber algebra structure in more details. The
approach is via representing the Hochschild complex as coderivations of the
tensor coalgebra of the underlying algebra, and representing the Gerstenhaber
bracket as a graded commutator of coderivations. A more general approach to computing the
Gerstenhaber bracket from an arbitrary resolution is presented.
Chapter 7 treats \(A_\infty\)- and \(L_\infty\)-algebras.
Chapter 8 is devoted to application of Hochschild cohomology to support
varieties of finite-dimensional algebras.
Chapter 9 treats Hopf algebras, their cohomology, and its relationship with
Hochschild cohomology.
The appendix contains necessary basic material from homological algebra.
The material illustrated by numerous examples, and augmented by numerous
exercises. The narrative is interspersed by interesting historical remarks, for
example, why ``bar resolvent'' is called ``bar'', etc.
The book compiles a lot of information from the literature in a convenient,
coherent, accurate, and entertaining manner; it is poised to become the standard
reference to the subject.
Of course, there is a lot of interesting material left (for example, Baer
invariants and connection with generators and relations, connections with
cohomology of Lie algebras, various spectral sequences, etc.), but for a book of
moderate size any choice is inevitable, and the choice here is very reasonable.Strict Mittag-Leffler modules and purely generated classeshttps://www.zbmath.org/1462.160032021-07-10T17:08:46.445117Z"Rothmaler, Philipp"https://www.zbmath.org/authors/?q=ai:rothmaler.philippSummary: We study versions of strict Mittag-Leffler modules relativized to a class \(\mathcal{K}\) (of modules), that is, \textit{strict} versions (in the technical sense of Raynaud and Gruson) of \(\mathcal{K}\)-Mittag-Leffler modules, as investigated in the preceding paper, \textit{Mittag-Leffler modules and definable subcategories}, in this very series.
For the entire collection see [Zbl 1437.18002].On a class of dual Rickart moduleshttps://www.zbmath.org/1462.160052021-07-10T17:08:46.445117Z"Tribak, R."https://www.zbmath.org/authors/?q=ai:tribak.rachidA module \(M\) is called \textsl{dual Rickart} (see [\textit{G. Lee} et al., Commun. Algebra 39, No. 11, 4036--4058 (2011; Zbl 1262.16005)]) if \(\mathrm{Im}f\) is a direct
summand of \(M\) for every \(f\in \mathrm{End}_{R}(M)\).
A module \(M\) is called \textsl{stable} (see [the reviewer and \textit{P. Schultz}, Bull. Aust. Math. Soc. 82, No. 1, 99--112 (2010; Zbl 1209.16005)]) if
all endomorphic images are fully invariant.
In this paper, the author investigates the intersection of the class of dual
Rickart modules with the class of stable modules.
An \(R\)-module \(M\) is called an \textsl{sd-Rickart} (also called \emph{strongly dual Rickart}) module if, for every nonzero endomorphism \(f\) of \(M\), \(\mathrm{Im}f\) is a fully invariant direct summand of \(M\). The main
results are the following
Theorem 2.1. Let a module \(M=\oplus _{i}M_{i}\) be a direct sum of
submodules \(M_{i}\), \(i\in I\). Then the following statements are equivalent:
(i) \(M\) is an sd-Rickart module;
(ii) \(M_{i}\) is an sd-Rickart submodule of \(M\) for every \(i\in I\) and \(%
\mathrm{Hom}_{R}(M_{i},M_{j})=0\) for all distinct \(i,j\in I\).
A module \(M\) is said to be \textsl{radical} if \(\mathrm{Rad}(M)=M\). The sum
of all radical submodules of a module \(M\) is denoted by \(P(M)\). If \(R\) is a
commutative Noetherian ring, then \(\mathrm{Ass}(M)\) denotes the set of all
prime ideals associated with \(M\).
Theorem 2.2. Let \(R\) be a commutative Noetherian ring and let \(
\Omega _{R}\) be the set of maximal ideals of \(R\). Also let \(M\) be an \(R\)
-module such that \(\mathrm{Ass}(M)\cap \Omega _{R}\) is a finite set. Then
the following conditions are equivalent:
(i) \(M\) is a dual Rickart \(R\)-module;
(ii) \(M=M_{1}\oplus M_{2}\) is a direct sum of a dual Rickart submodule \(M_{1}
\) and a semisimple submodule \(M_{2}\) such that \(M_{1}=\mathrm{Rad}(M_{1})=
\mathrm{Rad}(M)=P(M)\) and \(\mathrm{Hom}_{R}(M_{2},M_{1})=0\).
Proposition 2.9. Let \(M\) be a torsion-free module over a
commutative domain \(R\). If \(M\) is an sd-Rickart \(R\)-module, then \(M\) is
injective.
In addition, a generalization of sd-Rickart modules is studied, requiring,
for every nonzero endomorphism \(f\) of \(M\), that \(\mathrm{Im}f\) contains a
nonzero fully invariant direct summand of \(M\).Modules in which the annihilator of a fully invariant submodule is purehttps://www.zbmath.org/1462.160072021-07-10T17:08:46.445117Z"Amirzadeh Dana, P."https://www.zbmath.org/authors/?q=ai:dana.p-amirzadeh"Moussavi, A."https://www.zbmath.org/authors/?q=ai:moussavi.ahmadSummary: A ring \(R\) is called left \(AIP\) if \(R\) modulo the left annihilator of any ideal is flat. In this paper, we characterize a module \(M_R\) for which the endomorphism ring \(\mathrm{End}_R(M)\) is left \(AIP\). We say a module \(M_R\) is endo-\(AIP\) (resp. endo-\(APP\)) if \(M\) has the property that ``the left annihilator in \(\mathrm{End}_R(M)\) of every fully invariant submodule of \(M\) (resp. \(\mathrm{End}_R(M)m\), for every \(m \in M)\) is pure as a left ideal in \(\mathrm{End}_R(M)\)''. The notion of endo-\textit{AIP} (resp. endo-\(APP\)) modules generalizes the notion of Rickart and p.q.-Baer modules to a much larger class of modules. It is shown that every direct summand of an endo-\(AIP\) (resp. endo-\(APP\)) module inherits the property and that every projective module over a left \(AIP\) (resp. \(APP\))-ring is an endo-\(AIP\) (resp. endo-\(APP\)) module.The extension dimension of triangular matrix algebrashttps://www.zbmath.org/1462.160102021-07-10T17:08:46.445117Z"Zheng, Junling"https://www.zbmath.org/authors/?q=ai:zheng.junling"Gao, Hanpeng"https://www.zbmath.org/authors/?q=ai:gao.hanpengSummary: Let \(T, U\) be two Artin algebras and \(_UM_T\) be a \(U\)-\(T\)-bimodule. In this paper, we study the extension dimension of the formal triangular matrix algebra \(\Lambda=\begin{pmatrix} T & 0 \\ M & U \end{pmatrix}\). It is proved that if \(_UM,M_T\) are projective and \(\max\{\mathrm{gl}.\dim T,\dim\bmod U\}\geqslant 1\), then \(\max\{\dim \bmod T, \dim\bmod U\} \leqslant\dim\bmod\Lambda\leqslant\max\{\mathrm{gl}.\dim T,\dim\bmod U\}\).Quasi-hereditary covers of higher zigzag algebras of type \(A\)https://www.zbmath.org/1462.160152021-07-10T17:08:46.445117Z"Bocca, Gabriele"https://www.zbmath.org/authors/?q=ai:bocca.gabrieleIn this paper, the author generalizes the construction of quasi-hereditary covers to the case of higher zigzag algebras of type \(A\) and shows that they satisfy various Koszul properties. In particular, he shows that the results proved by \textit{D. O. Madsen} [J. Algebra 395, 96--110 (2013; Zbl 1290.16024)] about generalized Koszul duality apply for such algebras and computes explicitly their \(\Delta\)-Koszul dualRings whose singular ideals are nilhttps://www.zbmath.org/1462.160182021-07-10T17:08:46.445117Z"Ahmadi, M."https://www.zbmath.org/authors/?q=ai:ahmadi.morteza"Moussavi, A."https://www.zbmath.org/authors/?q=ai:moussavi.ahmadSummary: It is well known that when a ring \(R\) satisfies ACC on right annihilators of elements, then the right singular ideal of \(R\) is nil, in this case, we say \(R\) is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains \(\pi\)-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.Skew category algebrashttps://www.zbmath.org/1462.160192021-07-10T17:08:46.445117Z"Bavula, V. V."https://www.zbmath.org/authors/?q=ai:bavula.vladimir-vSummary: We study a new (large) class of algebras (that was introduced in our work [ibid. 11, No. 3--4, 253--268 (2017; Zbl 1381.16014)]) -- the \textit{skew category algebras}. Any such an algebra \(\mathcal{C}(\sigma)\) is constructed from a category \(\mathcal{C}\) and a functor \(\sigma\) from the category \(\mathcal{C}\) to the category of algebras. Criteria are given for the algebra \(\mathcal{C}(\sigma)\) to be simple or left Noetherian or right Noetherian or semiprime or have 1.On nil clean group ringshttps://www.zbmath.org/1462.160232021-07-10T17:08:46.445117Z"Cui, Jian"https://www.zbmath.org/authors/?q=ai:cui.jian"Li, Yuanlin"https://www.zbmath.org/authors/?q=ai:li.yuanlin"Wang, Haobai"https://www.zbmath.org/authors/?q=ai:wang.haobaiFor a ring \(R\) and a group \(G\), denote by \(RG\) the \textsl{group ring} of \(G\)
over \(R\). Results on nil-clean group rings include:
For a commutative ring \(R\) and an abelian group \(G\)
the group ring \(RG\) is nil clean if and only if \(R\) is nil
clean and \(G\) is a 2-group [\textit{W. Wm. McGovern} et al., J. Algebra Appl. 14, No. 6, Article ID 1550094, 5 p. (2015; Zbl 1325.16024)] or,
for a ring \(R\) and a symmetric group \(S_{3}\), the group
ring \(RS_{3}\) is nil clean if and only if both \(R\) and \(\mathbb{M}_{2}(R)\) are nil clean [\textit{S. Sahinkaya} et al., J. Algebra Appl. 16, No. 7, Article ID 1750135, 7 p. (2017; Zbl 1382.16021)].
In this paper, if \(D_{2n}\) denotes the \textsl{dihedral} group of order \(2n\)
and \(Q_{2n}\) denotes the \textsl{generalized quaternion} group of order \(2n\)
(with even \(n\)), the nil-cleanness of the group rings \(RD_{2n}\) and \(RQ_{2n}\)
are characterized.
The following result is proved.
Theorem 2.3 and 2.7. \(RD_{2n}\) is nil clean if and
only if \(RQ_{2n}\) is nil clean if and only if either \(n=2^{k}\)
and \(R\) is nil clean, or \(n=3\cdot 2^{k}\) and \(RS_{3}\)
is nil clean.
Finally, regarding nil \(\star \)-rings it is proved that
Theorem 3.4 If \(R\) is a commutative ring and \(G\)
is an abelian group then \(RG\) is nil \(\star \)-clean
if and only if both \(R\) and \(\mathbb{Z}_{2}G\) are nil clean.
Moreover,
Theorem 3.10. If \(R\) is a commutative ring, \(G\in
\{D_{2n},Q_{2n}\}\) (with even \(n\)), then \(RG\) is nil \(
\star \)-clean if and only if \(R\) is nil clean and \(G\)
is a 2-group.On idempotents of a class of commutative ringshttps://www.zbmath.org/1462.160242021-07-10T17:08:46.445117Z"de Melo Hernández, Fernanda D."https://www.zbmath.org/authors/?q=ai:de-melo-hernandez.fernanda-diniz"Hernández Melo, César A."https://www.zbmath.org/authors/?q=ai:hernandez-melo.cesar-adolfo"Tapia-Recillas, Horacio"https://www.zbmath.org/authors/?q=ai:tapia-recillas.horacioThe authors study idempotents of commutative rings starting from the classical result that if \(R\) is a ring with nilpotent ideal \(N\), then an idempotent of \(R/N\) can be lifted to an idempotent of \(R\). They introduce the following more general CNC-condition. A set \(\{N_1,\cdots,N_k \}\) of ideals in a commutative ring \(R\) satisfy the CNC-condition if: the ideals form a chain \(0 = N_k \subset N_{k-1} \subset \cdots \subset N_1 \subset R\) and for each \(1 \leq i \leq k-1\) there exist integers \(t_i\) and \(s_i\) such that \(N_i^t \subset N_{i+1}\) and \(s_iN_i \subset N_{i+1}\) and each prime factor of \(s_i\) is greater or equal to \(t_i\).
They show that under these conditions an idempotent \(f + N_1\) of \(R/N_1\) lifts to an idempotent \(f^{s_1s_2\cdots s_k}\) of \(R\). They then illustrate their result by applying it to several classes of rings with concrete examples, namely: commutative rings containing a nilpotent ideal and commutative group rings \(RG\) where \(R\) is a commutative ring containing a set of ideals satisfying the CNC-condition. Such rings \(R\) include chain rings and the residue rings \(\mathbb{Z}/m\mathbb{Z}\) where \(m\) is a positive integer.Semi-clean group ringshttps://www.zbmath.org/1462.160252021-07-10T17:08:46.445117Z"Klingler, L."https://www.zbmath.org/authors/?q=ai:klingler.lee-c"Loper, K. A."https://www.zbmath.org/authors/?q=ai:loper.k-alan"McGovern, W. Wm."https://www.zbmath.org/authors/?q=ai:mcgovern.warren-wm"Toeniskoetter, M."https://www.zbmath.org/authors/?q=ai:toeniskoetter.matthewSummary: Recall that a ring is said to be a \textit{clean ring} if every element can be expressed as the sum of a unit and an idempotent. In one variant of this definition, a ring is said to be a \textit{semi-clean ring} if every element can be expressed as the sum of a unit and a periodic element. \textit{Y. Ye}'s Theorem [Commun. Algebra 31, No. 11, 5609--5625 (2003; Zbl 1043.16015)] states that the group ring \(\mathbb{Z}_{(p)}[C_3]\) is semi-clean, where \(p\) is a prime integer and \(C_3\) is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if \(R\) is a local ring, then the group ring \(R [G]\) is semi-clean if and only if \(G\) is a torsion abelian group.\(G\)-invariant recollements and skew group algebrashttps://www.zbmath.org/1462.160262021-07-10T17:08:46.445117Z"Hu, Yonggang"https://www.zbmath.org/authors/?q=ai:hu.yonggang"Yao, Hailou"https://www.zbmath.org/authors/?q=ai:yao.hailouSummary: In this paper, we introduce the notion of \(G\)-invariant recollements. Moreover, a series of examples of \(G\)-invariant recollements are provided. It is shown that each \(G\)-invariant recollement of derived categories of algebras induces that of skew group algebras. Applying this reuslt, we show that \(K\)-group decompositions and singularity equivalences can be preserved by taking skew group algebras.Classification of simple modules of the Ore extension \(K[X][Y; f\frac{d}{dX}]\)https://www.zbmath.org/1462.160272021-07-10T17:08:46.445117Z"Bavula, V. V."https://www.zbmath.org/authors/?q=ai:bavula.vladimir-vSummary: For the algebras \(\Lambda\) in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of \(\Lambda\) are computed.Potential algebras with few generatorshttps://www.zbmath.org/1462.160282021-07-10T17:08:46.445117Z"Iyudu, Natalia"https://www.zbmath.org/authors/?q=ai:iyudu.natalia-k"Shkarin, Stanislav"https://www.zbmath.org/authors/?q=ai:shkarin.stanislav-aSummary: We give a complete description of quadratic twisted potential algebras on three generators as well as cubic twisted potential algebras on two generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic other than 2 or 3.Combinatorics and structure of Hecke-Kiselman algebrashttps://www.zbmath.org/1462.160292021-07-10T17:08:46.445117Z"Okniński, Jan"https://www.zbmath.org/authors/?q=ai:okninski.jan"Wiertel, Magdalena"https://www.zbmath.org/authors/?q=ai:wiertel.magdalenaAn introduction to a supersymmetric graph algebrahttps://www.zbmath.org/1462.160302021-07-10T17:08:46.445117Z"Radler, Katherine"https://www.zbmath.org/authors/?q=ai:radler.katherine"Srivastava, Ashish K."https://www.zbmath.org/authors/?q=ai:srivastava.ashish-kumarInjective stabilization of additive functors. II. (Co)torsion and the Auslander-Gruson-Jensen functorhttps://www.zbmath.org/1462.160312021-07-10T17:08:46.445117Z"Martsinkovsky, Alex"https://www.zbmath.org/authors/?q=ai:martsinkovsky.alex"Russell, Jeremy"https://www.zbmath.org/authors/?q=ai:russell.jeremyIn this paper, the second in a series of three, the authors define the torsion submodule of a module by using the injective stabilization of the tensor product with the module. If the underlying ring is a commutative domain, the torsion submodule coincides with the classical torsion submodule. Dually, by considering the projective stabilization of the Hom functor, the authors define the cotorsion quotient module of a module. They show some general properties of these two concepts. In particular, they consider the Auslander-Gruson-Jensen functor and its right adjoint and establish a duality between torsion and cotorsion over a ring with finitely presented injective envelope. The authors also present the derived functors of torsion and cotorsion under some finiteness conditions on the injective envelope of a ring.
For Part I, see [the authors, ibid. 530, 429--469 (2019; Zbl 1444.16009)].On Frobenius and separable Galois cowreathshttps://www.zbmath.org/1462.160342021-07-10T17:08:46.445117Z"Bulacu, D."https://www.zbmath.org/authors/?q=ai:bulacu.daniel"Torrecillas, B."https://www.zbmath.org/authors/?q=ai:torrecillas.blasIn this paper, the authors continue the study of Frobenius and separable cowreaths began in their previous paper [J. Noncommut. Geom. 9, 707--774 (2015; Zbl 1347.16035)], in the context of so-called \textit{Galois cowreaths}.
Let \(\mathcal C\) be a monoidal category with (co)equalizers (all of whose objects are assumed to be coflat and robust), and let \((A, X)\) be a Galois cowreath in \(\mathcal C\).
One of the main results of the paper establishes a one-to-one correspondence between Frobenius systems of the algebra extension \(A^{\mathrm{co}(X)} \to A\) in \(\mathcal C\) and Frobenius systems of a certain coalgebra \((X, \psi)\) associated to \((A, X)\). As a consequence of this it is shown that the extension \(A^{\mathrm{co}(X)} \to A\) is Frobenius if and only if the cowreath \((A, X)\) is Frobenius. Necessary and sufficient conditions
for a Frobenius pre-Galois cowreath to be Galois are also given.
Other results of the paper concern the notion of separability. In contrast with the previously mentioned result, the separability of the extension
\(A^{\mathrm{co}(X)} \to A\) is not equivalent to the separability of the cowreath \((A, X)\). The authors give here some necessary and sufficient conditions for the extension \(A^{\mathrm{co}(X)} \to A\) to be separable. They also show that a Frobenius Galois cowreath is separable if and only if it admits a total integral.Structure of NI rings related to centershttps://www.zbmath.org/1462.160372021-07-10T17:08:46.445117Z"Han, Juncheol"https://www.zbmath.org/authors/?q=ai:han.juncheol"Lee, Yang"https://www.zbmath.org/authors/?q=ai:lee.yang"Park, Sangwon"https://www.zbmath.org/authors/?q=ai:park.sangwonThe authors investigate the structure of so-called \textit{NI-rings} related to their center. A ring \(R\) is said to be an NI-ring, provided that the set \(\mathrm{Nil}(R)\) consisting of all nilpotent elements of \(R\) forms an ideal of \(R\). The ``pseudo'' version of these NI-rings is also considered. For instance, both NI-rings and pseudo NI-rings are Dedekind finite. Moreover, semisimple Artinian rings are pseudo NI-rings (see, Proposition 2.1 (1) and Corollary 2.2).
The paper under review is rather technical and so it is somewhat difficult to read. The main results are structured into four theorems -- e.g., Theorems 1.3, 1.4, 1.8, 2.3. Some clarifying remarks and examples are given as well, such as Remarks 1.2,1.5 and Examples 1.7,1.9.
An article relatively close to the paper under review and not cited in the bibliography is the following one: [\textit{J. Šter}, Carpathian J. Math. 32, No. 2, 251--258 (2016; Zbl 1399.16043)].Rings over which matrices are sums of idempotent and \(q \)-potent matriceshttps://www.zbmath.org/1462.160402021-07-10T17:08:46.445117Z"Abyzov, A. N."https://www.zbmath.org/authors/?q=ai:abyzov.adel-nailevich"Tapkin, D. T."https://www.zbmath.org/authors/?q=ai:tapkin.danil-tLet \(q\) be a natural and \(q>1\). An element \(r\) in a ring \(R\) is \(q\)\textsl{-potent} provided that \(r^{q}=r\).
In this paper, the rings over which each square matrix is the sum of an
idempotent matrix and a \(q\)-potent matrix are studied.
Using a series of ten preliminary lemmas, a first result is:
Theorem 11. If \(q\) is odd and such that one of the equivalences \(
q\equiv 1\) (mod 6) or \(q\equiv 1\) (mod 26) holds, then for every \(n\in
\mathbb{N}\) each matrix in \(\mathbb{M}_{n}(\mathbb{F}_{3})\) is the sum of an
idempotent and a \(q\)-potent.
Among the main results we mention:
Theorem 14. Let \(q>1\) be odd, and let \(R\) be an integral domain not
isomorphic to \(\mathbb{F}_{3}\). Then the following are equivalent:
(1) for every \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the
sum of an idempotent matrix and a \(q\)-potent matrix;
(2) for some \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the sum
of an idempotent matrix and a \(q\)-potent matrix;
(3) for every \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the
sum of a nilpotent matrix and a \(q\)-potent matrix;
(4) for some \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the sum
of a nilpotent matrix and a \(q\)-potent matrix;
(5) \(R\) is a finite field and \(\left\vert R\right\vert -1\) divides \(q-1\).
Theorem 22. Let \(q>1\) be odd, let \(R\) be a commutative ring, and
let \(\mathrm{Nil}(R)=0\). If \(R\) does not possess a homomorphic image isomorphic to \(
\mathbb{F}_{3}\), then the following are equivalent:
(1) for every \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the
sum of an idempotent matrix and a \(q\)-potent matrix;
(2) for some \(n\in \mathbb{N}\) each matrix in \(\mathbb{M}_{n}(R)\) is the sum
of an idempotent matrix and a \(q\)-potent matrix;
(3) the identity \(x^{q}=x\) holds in \(R\).
Theorem 23. Let \(R\) be a commutative ring, and \(\mathrm{Nil}(R)=0\). Assume
that one of the equalities \(q=6k+1\) and \(q=26k+1\) holds, where \(k\in \mathbb{
N}\). Then the following are equivalent:
(1) for all \(n\in N\) every matrix in \(\mathbb{M}_{n}(R)\) is the sum of a \(q\)-potent matrix and an idempotent matrix;
(2) for some \(n\in N\) every matrix in \(\mathbb{M}_{n}(R)\) is the sum of a \(q\)-potent matrix and an idempotent matrix;
(3) the identity \(x^{q}=x\) holds in \(R\).
Many examples and counterexamples are given.Some algorithmic problems for Poisson algebrashttps://www.zbmath.org/1462.170242021-07-10T17:08:46.445117Z"Zhang, Zerui"https://www.zbmath.org/authors/?q=ai:zhang.zerui"Chen, Yuqun"https://www.zbmath.org/authors/?q=ai:chen.yuqun"Bokut, Leonid A."https://www.zbmath.org/authors/?q=ai:bokut.leonid-aSummary: We develop a new theory of Gröbner-Shirshov bases for Poisson algebras, which requires to introduce several nontrivial new techniques. As applications, we solve the word problem for finitely presented Poisson algebras in the case of a single relation with a leading monomial of Lie type and in the case when enough relations involve the Poisson bracket of generators.Jellyfish partition categorieshttps://www.zbmath.org/1462.180042021-07-10T17:08:46.445117Z"Comes, Jonathan"https://www.zbmath.org/authors/?q=ai:comes.jonathanFor each positive integer \(n\), the author introduces a monoidal category \(\mathcal{JP}(n)\), called the jellyfish partition category, using a generalisation of partition diagrams. This category is equipped with a monoidal functor \(\Psi:\mathcal{JP}(n)\to\operatorname{Rep}(A_n)\), where \(A_n\) is the alternating group on \(n\) letters. The main results are that \(\Psi\) is always full, and that, if \(n>1\), and either the characteristic of the field is zero or greater or equal to \(n\), then \(\Psi\) is faithful. Hence, \(\mathcal{JP}(n)\) is monoidally equivalent to the full subcategory of \(\operatorname{Rep}(A_n)\) whose objects are tensor powers of the natural \(n\)-dimensional permutation representation.Golden mean renormalization for the almost Mathieu operator and related skew productshttps://www.zbmath.org/1462.810912021-07-10T17:08:46.445117Z"Koch, Hans"https://www.zbmath.org/authors/?q=ai:koch.hans-friedrichSummary: Considering \(\text{SL}(2, \mathbb{R})\) skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean \(\alpha_*\) has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence that this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at \(\alpha_*\) and scaling properties of the generalized eigenfunction for this energy.
{\copyright 2021 American Institute of Physics}