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Publications (10 of 38) Show all publications
Nystedt, P., Öinert, J. & Richter, J. (2019). Simplicity of Ore monoid rings. Journal of Algebra, 530, 69-85
Open this publication in new window or tab >>Simplicity of Ore monoid rings
2019 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 530, p. 69-85Article in journal (Refereed) Published
Abstract [en]

Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring R[π;G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids. © 2019 Elsevier Inc.

National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-13849 (URN)10.1016/j.jalgebra.2019.04.003 (DOI)000469166400003 ()2-s2.0-85064169587 (Scopus ID)
Available from: 2019-05-24 Created: 2019-05-24 Last updated: 2019-07-24Bibliographically approved
Nystedt, P., Öinert, J. & Pinedo, H. (2018). Artinian and noetherian partial skew groupoid rings. Journal of Algebra, 503, 433-452
Open this publication in new window or tab >>Artinian and noetherian partial skew groupoid rings
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, p. 433-452Article in journal (Refereed) Published
Abstract [en]

Let alpha = {alpha(g) : Rg-1 -> R-g}(g is an element of mor(G)) be a partial action of a groupoid G on a (not necessarily associative) ring R and let S = R-star alpha G be the associated partial skew groupoid ring. We show that if a is global and unital, then S is left (right) artinian if and only if R is left (right) artinian and R-g = {0}, for all but finitely many g is an element of mor(G). We use this result to prove that if a is unital and R is alternative, then S is left (right) artinian if and only if R is left (right) artinian and R-g = {0}, for all but finitely many g is an element of mor(G). This result applies to partial skew group rings, in particular. Both of the above results generalize a theorem by J. K. Park for classical skew group rings, i.e. the case when R is unital and associative, and G is a group which acts globally on R. We provide two additional applications of our main results. Firstly, we generalize I. G. Connell's classical result for group rings by giving a characterization of artinian (not necessarily associative) groupoid rings. This result is in turn applied to partial group algebras. Secondly, we give a characterization of artinian Leavitt path algebras. At the end of the article, we relate noetherian and artinian properties of partial skew groupoid rings to those of global skew groupoid rings, as well as establish two Maschke-type results, thereby generalizing results by M. Ferrero and J. Lazzarin for partial skew group rings to the case of partial skew groupoid rings.

National Category
Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-12244 (URN)10.1016/j.jalgebra.2018.02.007 (DOI)000429764400020 ()2-s2.0-85044284762 (Scopus ID)
Note

Available online 14 February 2018

Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2019-05-28Bibliographically approved
Nystedt, P., Öinert, J. & Pinedo, H. (2018). Epsilon-strongly graded rings, separability and semisimplicity. Journal of Algebra, 514, 1-24
Open this publication in new window or tab >>Epsilon-strongly graded rings, separability and semisimplicity
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, p. 1-24Article in journal (Refereed) Published
Abstract [en]

We introduce the class of epsilon-strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products. We determine precisely when an epsilon-strongly graded ring is separable over its principal component. Thereby, we simultaneously generalize a result for strongly group graded rings by Nǎstǎsescu, Van den Bergh and Van Oystaeyen, and a result for unital partial crossed products by Bagio, Lazzarin and Paques. We also show that the class of unital partial crossed products appears in the class of epsilon-strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings. Thereby, we obtain, in the special case of unital partial crossed products, a short proof of a general result by Dokuchaev, Exel and Simón concerning when graded rings can be presented as partial crossed products. We also provide some interesting classes of examples of separable epsilon-strongly graded rings, with finite as well as infinite grading groups. In particular, we obtain an answer to a question raised by Le Bruyn, Van den Bergh and Van Oystaeyen in 1988. © 2018 Elsevier Inc.

Keywords
Group graded ring, Partial crossed product, SeparableSemisimple, Frobenius
National Category
Mathematics Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-12978 (URN)10.1016/j.jalgebra.2018.08.002 (DOI)000445848900001 ()2-s2.0-85051634856 (Scopus ID)
Available from: 2018-10-29 Created: 2018-10-29 Last updated: 2019-05-28Bibliographically approved
Nystedt, P., Öinert, J. & Richter, J. (2018). Non-associative Ore extensions. Israel Journal of Mathematics, 224(1), 263-292
Open this publication in new window or tab >>Non-associative Ore extensions
2018 (English)In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 224, no 1, p. 263-292Article in journal (Refereed) Published
Abstract [en]

We introduce non-associative Ore extensions, S = R[X; sigma, delta], for any non-ssociative unital ring R and any additive maps sigma, delta : R -> R satisfying sigma(1) = 1 and delta(1) = 0. In the special case when delta is either left or right R-delta-linear, where R-delta = ker(delta), and R is delta-simple, i.e. 0 and R are the only delta-invariant ideals of R, we determine the ideal structure of the non-associative differential polynomial ring D = R[X; id(R),delta]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z(D) = R-delta[p] for a monic p is an element of R-delta [X], unique up to addition of elements from Z(R)(delta) . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is delta-simple and Z(D) equals the field R-delta boolean AND Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silve-strov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R-delta boolean AND Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.

National Category
Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-12481 (URN)10.1007/s11856-018-1642-z (DOI)000431796000010 ()2-s2.0-85044256972 (Scopus ID)
Note

First Online: 06 March 2018

Available from: 2018-06-15 Created: 2018-06-15 Last updated: 2019-05-28Bibliographically approved
Nystedt, P. (2018). Partial category actions on sets and topological spaces. Communications in Algebra, 46(2), 671-683
Open this publication in new window or tab >>Partial category actions on sets and topological spaces
2018 (English)In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 46, no 2, p. 671-683Article in journal (Refereed) Published
Abstract [en]

We introduce (continuous) partial category actions on sets (topological spaces) and show that each such action admits a universal globalization. Thereby, we obtain a simultaneous generalization of corresponding results for groups, by Abadie, and Kellendonk and Lawson, and for monoids, by Megrelishvili and Schroder. We apply this result to the special case of partial groupoid actions where we obtain a sharpening of a result by Gilbert, concerning ordered groupoids, in the sense that mediating functions between universal globalizations always are injective.

Keywords
Globalization; partial group action; partial groupoid action
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-13484 (URN)10.1080/00927872.2017.1327057 (DOI)000418083100016 ()2-s2.0-85020741297 (Scopus ID)
Available from: 2019-02-06 Created: 2019-02-06 Last updated: 2019-05-28Bibliographically approved
Nystedt, P. (2017). A proof of the law of sines using the law of cosines. Mathematics Magazine, 90(3), 180-181
Open this publication in new window or tab >>A proof of the law of sines using the law of cosines
2017 (English)In: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 90, no 3, p. 180-181Article in journal (Refereed) Published
Abstract [en]

We give a proof of the law of sines using the law of cosines. © Mathematical Association of America.

National Category
Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-11914 (URN)10.4169/math.mag.90.3.180 (DOI)2-s2.0-85032820611 (Scopus ID)
Available from: 2017-12-12 Created: 2017-12-12 Last updated: 2019-05-23Bibliographically approved
Nystedt, P. (2015). Fuzzy crossed product algebras. Annals of Fuzzy Mathematics and Informatics, 10(6), 959-969
Open this publication in new window or tab >>Fuzzy crossed product algebras
2015 (English)In: Annals of Fuzzy Mathematics and Informatics, ISSN 2093-9310, E-ISSN 2287-6235, Vol. 10, no 6, p. 959-969Article in journal (Refereed) Published
Abstract [en]

We introduce fuzzy groupoid graded rings and, as a par-ticular case, fuzzy crossed product algebras. We show that there is abijection between the set of fuzzy graded is omorphism equivalence classes of fuzzy crossed product algebras and the associated second cohomology group. This generalizes a classical result for crossed product algebras to thefuzzy situation. Thereby, we quantize the difference of richness between the fuzzy and the crisp case. We give several examples showing that in the fuzzy case the associated second cohomology group is much ner than in the classical situation. In particular, we show that the cohomology group may by in nite in the fuzzy case even though it is trivial in the crisp case.

Keywords
Fuzzy ring, Fuzzy group, Cohomolgy
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-8911 (URN)
Available from: 2016-01-21 Created: 2016-01-21 Last updated: 2019-05-13Bibliographically approved
Nystedt, P. & Öinert, J. (2015). Outer Partial Actions and Partial Skew Group Rings. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 67(5), 1144-1160
Open this publication in new window or tab >>Outer Partial Actions and Partial Skew Group Rings
2015 (English)In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 67, no 5, p. 1144-1160Article in journal (Refereed) Published
Abstract [en]

We extend the classicial notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A⋆αG is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Keywords
Outer action, partial action, minimality, topological dynamics, partial skew group ring, simplicity
National Category
Natural Sciences
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-8010 (URN)10.4153/CJM-2014-043-8 (DOI)000362152600008 ()2-s2.0-84976400598 (Scopus ID)
Available from: 2015-08-27 Created: 2015-08-27 Last updated: 2019-05-14Bibliographically approved
Nystedt, P. & Oinert, J. (2015). Simple semigroup graded rings. Journal of Algebra and its Applications, 14(7)
Open this publication in new window or tab >>Simple semigroup graded rings
2015 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 14, no 7Article in journal (Refereed) Published
Abstract [en]

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and R-e has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring fR(eGe)f is a field. This is a generalization of a result of Jespers’ on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Goncalves’. We also point out how Jespers’ result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

Keywords
Semigroup graded ring, partial skew group ring, simplicity
National Category
Mathematics
Identifiers
urn:nbn:se:hv:diva-7623 (URN)10.1142/S0219498815501029 (DOI)000353552200006 ()2-s2.0-84928555107 (Scopus ID)
Available from: 2015-05-30 Created: 2015-05-30 Last updated: 2017-12-04Bibliographically approved
Nystedt, P. (2014). A Proof of the Cosine Addition Formula Using the Law of Cosines. Mathematics Magazine, 87(2), 144-144
Open this publication in new window or tab >>A Proof of the Cosine Addition Formula Using the Law of Cosines
2014 (English)In: Mathematics Magazine, ISSN 0025-570X, E-ISSN 1930-0980, Vol. 87, no 2, p. 144-144Article in journal (Refereed) Published
Abstract [en]

We give a proof of the cosine addition formula using the law of cosines.

National Category
Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-6223 (URN)10.4169/math.mag.87.2.144 (DOI)2-s2.0-84903604672 (Scopus ID)
Available from: 2014-05-08 Created: 2014-05-08 Last updated: 2019-05-07Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0001-6594-7041

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