Recent zbMATH articles in MSC 16S35https://www.zbmath.org/atom/cc/16S352021-07-10T17:08:46.445117ZWerkzeugLaurent 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.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.\(G\)-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.