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Öinert, Johan, ProfessorORCID iD iconorcid.org/0000-0001-8095-0820
Alternative names
Publications (10 of 27) Show all publications
Machado, N., Öinert, J. & Wagner, S. (2025). Non-Abelian Extensions of Groupoids and Their Groupoid Rings. Applied Categorical Structures, 33(1), Article ID 5.
Open this publication in new window or tab >>Non-Abelian Extensions of Groupoids and Their Groupoid Rings
2025 (English)In: Applied Categorical Structures, ISSN 0927-2852, E-ISSN 1572-9095, Vol. 33, no 1, article id 5Article in journal (Refereed) Published
Abstract [en]

We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids N→E→G gives rise to a groupoid crossed product of G by the groupoid ring of N which recovers the groupoid ring of E up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products.

Place, publisher, year, edition, pages
Springer Science+Business Media B.V., 2025
Keywords
Non-Abelian extension of groupoids, factor system, groupoid cohomology, groupoid crossed product, groupoid ring, groupoid C^∗-algebra
National Category
Algebra and Logic Geometry
Research subject
Mathematics and applications
Identifiers
urn:nbn:se:bth-26318 (URN)10.1007/s10485-024-09795-8 (DOI)001379547600001 ()2-s2.0-85212061459 (Scopus ID)
Available from: 2024-06-03 Created: 2024-06-03 Last updated: 2025-01-02Bibliographically approved
Lännström, D., Lundström, P., Öinert, J. & Wagner, S. (2025). Prime group graded rings with applications to partial crossed products and Leavitt path algebras. Journal of Pure and Applied Algebra, 229(1), Article ID 107842.
Open this publication in new window or tab >>Prime group graded rings with applications to partial crossed products and Leavitt path algebras
2025 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 229, no 1, article id 107842Article in journal (Refereed) Published
Abstract [en]

We generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a characterization of prime s-unital strongly group graded rings, and, in particular, of infinite matrix rings and of group rings over s-unital rings, thereby generalizing a well-known result by Connell; (ii) characterizations of prime s-unital partial skew group rings and of prime unital partial crossed products; (iii) a generalization of the well-known characterizations of prime Leavitt path algebras, by Larki and by Abrams-Bell-Rangaswamy. © 2024 The Author(s)

Place, publisher, year, edition, pages
Elsevier, 2025
Keywords
Group graded ring, Leavitt path algebra, Nearly epsilon-strongly graded ring, Partial skew group ring, Prime ring, Unital partial crossed product
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-27256 (URN)10.1016/j.jpaa.2024.107842 (DOI)001372913300001 ()2-s2.0-85210667645 (Scopus ID)
Available from: 2024-12-17 Created: 2024-12-17 Last updated: 2024-12-27Bibliographically approved
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
Lännström, D. & Öinert, J. (2024). Graded von Neumann regularity of rings graded by semigroups. Beitraege zur Algebra und Geometrie, 65(1), 13-21
Open this publication in new window or tab >>Graded von Neumann regularity of rings graded by semigroups
2024 (English)In: Beitraege zur Algebra und Geometrie, ISSN 0138-4821, E-ISSN 2191-0383, Vol. 65, no 1, p. 13-21Article in journal (Refereed) Published
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, 2024
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: 2024-06-24Bibliographically approved
Moreira, P. S. E. & Öinert, J. (2024). Prime groupoid graded rings with applications to partial skew groupoid rings. Communications in Algebra, 52(7), 3134-3153
Open this publication in new window or tab >>Prime groupoid graded rings with applications to partial skew groupoid rings
2024 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 52, no 7, p. 3134-3153Article in journal (Refereed) Published
Abstract [en]

In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg. © 2024 The Author(s). Published with license by Taylor & Francis Group, LLC.

Place, publisher, year, edition, pages
Taylor & Francis, 2024
Keywords
Group-type partial action, groupoid graded ring, nearly epsilon-strongly groupoid graded ring, partial skew groupoid ring, prime ring
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-26059 (URN)10.1080/00927872.2024.2315311 (DOI)001174001000001 ()2-s2.0-85186872447 (Scopus ID)
Available from: 2024-03-19 Created: 2024-03-19 Last updated: 2024-06-24Bibliographically 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, 36(4), 1081-1117
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 1435-5337, Vol. 36, no 4, p. 1081-1117Article in journal (Refereed) Published
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: 2024-08-05Bibliographically approved
Öinert, J. (2024). Units, zero-divisors and idempotents in rings graded by torsion-free groups. Journal of Group Theory, 27(4), 789-811
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 Theory, ISSN 1433-5883, E-ISSN 1435-4446, Vol. 27, no 4, p. 789-811Article in journal (Refereed) Published
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: 2024-08-05Bibliographically approved
Lundström, P., Öinert, J., Orozco, L. & Pinedo, H. (2024). Very good gradings on matrix rings are epsilon-strong. Linear and multilinear algebra
Open this publication in new window or tab >>Very good gradings on matrix rings are epsilon-strong
2024 (English)In: Linear and multilinear algebra, ISSN 0308-1087, E-ISSN 1563-5139Article in journal (Refereed) Epub ahead of print
Abstract [en]

We investigate properties of group gradings on matrix rings (Formula presented.), where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on (Formula presented.) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that (Formula presented.) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when (Formula presented.) is an epsilon-crossed product. Our results are illustrated by several examples. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Place, publisher, year, edition, pages
Taylor & Francis, 2024
Keywords
epsilon-strongly graded ring, good grading, Matrix ring, unital partial crossed product, very good grading
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-26175 (URN)10.1080/03081087.2024.2314205 (DOI)001206069700001 ()2-s2.0-85191188101 (Scopus ID)
Available from: 2024-05-07 Created: 2024-05-07 Last updated: 2024-05-07Bibliographically 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
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-8095-0820

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