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Publications (10 of 38) Show all publications
Nystedt, P. & Öinert, J. (2019). Group gradations on Leavitt path algebras. Journal of Algebra and its Applications
Open this publication in new window or tab >>Group gradations on Leavitt path algebras
2019 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829Article in journal (Refereed) Epub ahead of print
Abstract [en]

Given a directed graph E and an associative unital ring R one may define the Leavitt path algebra with coefficients in R, denoted by LR(E). For an arbitrary group G, LR(E) can be viewed as a G-graded ring. In this paper, we show that LR(E) is always nearly epsilon-strongly G-graded. We also show that if E is finite, then LR(E) is epsilon-strongly G-graded. We present a new proof of Hazrat’s characterization of strongly g-graded Leavitt path algebras, when E is finite. Moreover, if E is row-finite and has no source, then we show that LR(E) is strongly-graded if and only if E has no sink. We also use a result concerning Frobenius epsilon-strongly G-graded rings, where G is finite, to obtain criteria which ensure that LR(E) is Frobenius over its identity component. © 2020 World Scientific Publishing Company.

Keywords
s -unital ring, strongly graded ring, epsilon-strongly graded ring, Leavitt path algebra
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-14470 (URN)10.1142/S0219498820501650 (DOI)2-s2.0-85071375607 (Scopus ID)
Available from: 2019-10-02 Created: 2019-10-02 Last updated: 2020-01-17Bibliographically approved
Nystedt, P. (2019). Poisson’s fundamental theorem of calculus via Taylor’s formula. International Journal of Mathematical Education in Science and Technology
Open this publication in new window or tab >>Poisson’s fundamental theorem of calculus via Taylor’s formula
2019 (English)In: International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, E-ISSN 1464-5211Article in journal (Refereed) Epub ahead of print
Abstract [en]

We use Taylor’s formula with Lagrange remainder to make a modern adaptation of Poisson’s proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

Keywords
Reimann sums, Taylor´s formula, the fundamental theorem of calculus
National Category
Other Mathematics
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-14871 (URN)10.1080/0020739X.2019.1682699 (DOI)000494895600001 ()2-s2.0-85074977999 (Scopus ID)
Available from: 2020-01-29 Created: 2020-01-29 Last updated: 2020-01-29Bibliographically approved
Nystedt, P. & Öinert, J. (2019). Simple graded rings, nonassociative crossed products and Cayley-Dickson doublings. Journal of Algebra and its Applications, Article ID 2050231.
Open this publication in new window or tab >>Simple graded rings, nonassociative crossed products and Cayley-Dickson doublings
2019 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, article id 2050231Article in journal (Refereed) Epub ahead of print
Abstract [en]

e show that if a nonassociative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a nonassociative setting. By applying this result to nonassociative crossed products, we obtain nonassociative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon. © 2020 World Scientific Publishing Company.

Keywords
Cayley algebra, group graded ring, nonassociative crossed product, Nonassociative ring, simplicity
National Category
Algebra and Logic
Research subject
ENGINEERING, Mathematics
Identifiers
urn:nbn:se:hv:diva-14872 (URN)10.1142/S021949882050231X (DOI)2-s2.0-85076527806 (Scopus ID)
Available from: 2020-01-29 Created: 2020-01-29 Last updated: 2020-01-29Bibliographically approved
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: 2020-02-03Bibliographically 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
Algebra and Logic
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: 2019-10-24Bibliographically 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
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: 2019-10-24Bibliographically 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
Algebra and Logic
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: 2019-10-24Bibliographically 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
Geometry
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: 2019-12-05Bibliographically 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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-6594-7041

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