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Öinert, Johan, ProfessorORCID iD iconorcid.org/0000-0001-8095-0820
Alternative names
Publications (10 of 21) Show all publications
Lorensen, K. & Öinert, J. (2024). Generating numbers of rings graded by amenable and supramenable groups. Journal of the London Mathematical Society, 109(1), Article ID e12826.
Open this publication in new window or tab >>Generating numbers of rings graded by amenable and supramenable groups
2024 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 109, no 1, article id e12826Article in journal (Refereed) Published
Abstract [en]

A ring 𝑅 has unbounded generating number (UGN) if,for every positive integer 𝑛, there is no 𝑅-module epimorphism 𝑅𝑛 → 𝑅𝑛+1. For a ring 𝑅 = ⨁g∈𝐺 𝑅g gradedby a group 𝐺 such that the base ring 𝑅1 has UGN, weidentify several sets of conditions under which 𝑅 mustalso have UGN. The most important of these are: (1)𝐺 is amenable, and there is a positive integer 𝑟 suchthat, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as 𝑅1-modules for some𝑖 = 1, … , 𝑟; (2) 𝐺 is supramenable, and there is a positive integer 𝑟 such that, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as𝑅1-modules for some 𝑖 = 0, … , 𝑟. The pair of conditions(1) leads to three different ring-theoretic characterizations of the property of amenability for groups. We alsoconsider rings that do not have UGN; for such a ring𝑅, the smallest positive integer 𝑛 such that there is an𝑅-module epimorphism 𝑅𝑛 → 𝑅𝑛+1 is called the generating number of 𝑅, denoted gn(𝑅). If 𝑅 has UGN, then wedefine gn(𝑅) ∶= ℵ0. We describe several classes of examples of a ring 𝑅 graded by an amenable group 𝐺 such thatgn(𝑅) ≠ gn(𝑅1).

MSC 2020

16P99, 16S35, 16W50, 20F65, 43A07 (primary), 16D90 (secondary)

Place, publisher, year, edition, pages
John Wiley & Sons, 2024
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-25553 (URN)10.1112/jlms.12826 (DOI)001119522700001 ()2-s2.0-85174638607 (Scopus ID)
Available from: 2023-11-07 Created: 2023-11-07 Last updated: 2023-12-31Bibliographically approved
Bagio, D., Gonçalves, D., Moreira, P. S. & Öinert, J. (2024). The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras. Forum mathematicum
Open this publication in new window or tab >>The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras
2024 (English)In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337Article in journal (Refereed) Epub ahead of print
Abstract [en]

Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R ⋊ α G {R\rtimes_{\alpha}G}, which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R ⋊ α G {R\rtimes_{\alpha}G}. We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G {R\rtimes_{\alpha}G}. We show that every ideal of R ⋊ α G {R\rtimes_{\alpha}G} is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra. © 2024 Walter de Gruyter GmbH, Berlin/Boston 2024.

Place, publisher, year, edition, pages
Walter de Gruyter, 2024
Keywords
condition (K), graded ideal, Partial skew groupoid ring, residual intersection property, topological freeness, topological transitivity, ultragraph algebra
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-25888 (URN)10.1515/forum-2023-0117 (DOI)001134564900001 ()2-s2.0-85181448378 (Scopus ID)
Available from: 2024-01-12 Created: 2024-01-12 Last updated: 2024-02-01Bibliographically approved
Öinert, J. (2024). Units, zero-divisors and idempotents in rings graded by torsion-free groups. Journal of group theroy
Open this publication in new window or tab >>Units, zero-divisors and idempotents in rings graded by torsion-free groups
2024 (English)In: Journal of group theroy, ISSN 1433-5883, E-ISSN 1435-4446Article in journal (Refereed) Epub ahead of print
Abstract [en]

The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

Place, publisher, year, edition, pages
Walter de Gruyter, 2024
Keywords
group graded ring, torsion-free group, unique product group, unit conjecture, zero-divisor conjecture, idempotent conjecture
National Category
Algebra and Logic
Research subject
Mathematics and applications
Identifiers
urn:nbn:se:bth-21688 (URN)10.1515/jgth-2023-0110 (DOI)001151748200001 ()2-s2.0-85184045564 (Scopus ID)
Available from: 2021-06-17 Created: 2021-06-17 Last updated: 2024-02-16Bibliographically approved
Öinert, J. & Wagner, S. (2023). Complex group rings and group C ∗ -algebras of group extensions. Journal of Algebraic Combinatorics, 58(2), 387-397
Open this publication in new window or tab >>Complex group rings and group C ∗ -algebras of group extensions
2023 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 58, no 2, p. 387-397Article in journal (Refereed) Published
Abstract [en]

Let N and H be groups, and let G be an extension of H by N. In this article, we describe the structure of the complex group ring of G in terms of data associated with N and H. In particular, we present conditions on the building blocks N and H guaranteeing that G satisfies the zero-divisor and idempotent conjectures. Moreover, for central extensions involving amenable groups we present conditions on the building blocks guaranteeing that the Kadison–Kaplansky conjecture holds for the group C∗-algebra of G. © 2022, The Author(s).

Place, publisher, year, edition, pages
Springer, 2023
Keywords
Complex group ring, Crossed product, Crossed system, Group C∗-algebra, Group extension, Idempotent conjecture, Kadison–Kaplansky conjecture, Torsion-free group, Zero-divisor conjecture
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-23829 (URN)10.1007/s10801-022-01183-6 (DOI)000871156400001 ()2-s2.0-85140467387 (Scopus ID)
Funder
Carl Tryggers foundation , 16:540Carl Tryggers foundation , KF17:27
Note

open access

Available from: 2022-11-04 Created: 2022-11-04 Last updated: 2023-12-05Bibliographically approved
Lundstrom, P., Öinert, J. & Richter, J. (2023). NON-UNITAL ORE EXTENSIONS. Colloquium Mathematicum, 172(2), 217-229
Open this publication in new window or tab >>NON-UNITAL ORE EXTENSIONS
2023 (English)In: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 172, no 2, p. 217-229Article in journal (Refereed) Published
Abstract [en]

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings R[x; delta], under the hy-pothesis that R is s-unital and ker(delta) contains a non-zero idempotent. This result gener-alizes a result by oinert, Richter and Silvestrov from the unital setting. We also present a family of examples of simple non-unital differential polynomial rings.

Place, publisher, year, edition, pages
Polish Academy of Sciences, 2023
Keywords
non-unital ring, Ore extension, simple ring, outer derivation
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-24330 (URN)10.4064/cm8941-11-2022 (DOI)000917921400001 ()2-s2.0-85163993902 (Scopus ID)
Available from: 2023-03-02 Created: 2023-03-02 Last updated: 2023-08-07Bibliographically approved
Baghdari, S. & Öinert, J. (2023). Pure semisimple and Kothe group rings. Communications in Algebra, 51(7), 2779-2790
Open this publication in new window or tab >>Pure semisimple and Kothe group rings
2023 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 51, no 7, p. 2779-2790Article in journal (Refereed) Published
Abstract [en]

In this article, we provide a complete characterization of abelian group rings which are Kothe rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are Kothe rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples.Communicated by Eric Jespers

Place, publisher, year, edition, pages
Taylor & Francis, 2023
Keywords
Group ring, Kothe ring, pure semisimple
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-24413 (URN)10.1080/00927872.2023.2172179 (DOI)000948849600001 ()2-s2.0-85150688860 (Scopus ID)
Available from: 2023-04-05 Created: 2023-04-05 Last updated: 2023-06-19Bibliographically approved
Lundstrom, P. & Öinert, J. (2023). SIMPLICITY OF LEAVITT PATH ALGEBRAS VIA GRADED RING THEORY. Bulletin of the Australian Mathematical Society, 108(3), 428-437
Open this publication in new window or tab >>SIMPLICITY OF LEAVITT PATH ALGEBRAS VIA GRADED RING THEORY
2023 (English)In: Bulletin of the Australian Mathematical Society, ISSN 0004-9727, E-ISSN 1755-1633, Vol. 108, no 3, p. 428-437Article in journal (Refereed) Published
Abstract [en]

Suppose that R is an associative unital ring and that E= (E-0, E-1, r, s) is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra L-R(E) is simple if and only if R is simple, E-0 has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

Place, publisher, year, edition, pages
Cambridge University Press, 2023
Keywords
Leavitt path algebra, graded ring, simple ring, centre
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-24412 (URN)10.1017/S0004972723000114 (DOI)000943276000001 ()2-s2.0-85177820296 (Scopus ID)
Available from: 2023-04-05 Created: 2023-04-05 Last updated: 2023-12-08Bibliographically approved
Lännström, D. & Öinert, J. (2022). Graded von Neumann regularity of rings graded by semigroups. Beitraege zur Algebra und Geometrie
Open this publication in new window or tab >>Graded von Neumann regularity of rings graded by semigroups
2022 (English)In: Beitraege zur Algebra und Geometrie, ISSN 0138-4821, E-ISSN 2191-0383Article in journal (Refereed) Epub ahead of print
Abstract [en]

In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded rings. © 2022, The Author(s).

Place, publisher, year, edition, pages
Springer, 2022
Keywords
Graded ring, Groupoid, Regular ring, Semigroup, von Neumann regular ring
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-24021 (URN)10.1007/s13366-022-00673-9 (DOI)000886883200001 ()2-s2.0-85142449190 (Scopus ID)
Note

open access

Available from: 2022-12-02 Created: 2022-12-02 Last updated: 2022-12-12Bibliographically approved
Lundström, P. & Öinert, J. (2022). Strongly graded leavitt path algebras. Journal of Algebra and its Applications, 21(07), Article ID 2250141.
Open this publication in new window or tab >>Strongly graded leavitt path algebras
2022 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 21, no 07, article id 2250141Article in journal (Refereed) Published
Abstract [en]

Let R be a unital ring, let E be a directed graph and recall that the Leavitt path algebra LR(E) carries a natural-gradation. We show that LR(E) is strongly-graded if and only if E is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained. © 2022 World Scientific Publishing Company

Place, publisher, year, edition, pages
World Scientific, 2022
Keywords
Leavitt path algebra, Strongly graded ring
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-21398 (URN)10.1142/S0219498822501419 (DOI)000820925200009 ()2-s2.0-85104961993 (Scopus ID)
Available from: 2021-05-17 Created: 2021-05-17 Last updated: 2022-08-11Bibliographically approved
Nystedt, P., Öinert, J. & Pinedo, H. (2020). EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY. Glasgow Mathematical Journal, 62(1), 233-259, Article ID PII S0017089519000065.
Open this publication in new window or tab >>EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY
2020 (English)In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 62, no 1, p. 233-259, article id PII S0017089519000065Article in journal (Refereed) Published
Abstract [en]

We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.

Place, publisher, year, edition, pages
CAMBRIDGE UNIV PRESS, 2020
Keywords
Primary: 16W50, Secondary: 16E99, 16D99, 14C22
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-19023 (URN)10.1017/S0017089519000065 (DOI)000500321900015 ()
Note

open access

Available from: 2019-12-18 Created: 2019-12-18 Last updated: 2019-12-27Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-8095-0820

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