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Nystedt, Patrikorcid.org/0000-0001-6594-7041

Open this publication in new window or tab >>Simplicity of Ore monoid rings### Nystedt, Patrik

### Öinert, Johan

### Richter, Johan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 530, p. 69-85Article in journal (Refereed) Published
##### Abstract [en]

##### 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)
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Available from: 2019-05-24 Created: 2019-05-24 Last updated: 2019-07-24Bibliographically approved

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.

Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.

Mälardalen University, Academy of Education, Culture and Communication,Box 883, Västerås, SE-72123, Sweden.

Given a non-associative unital ring R, a monoid G and a set Ï of additive maps RâR, we introduce the Ore monoid ring R[Ï;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 Ï. 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.

Open this publication in new window or tab >>Artinian and noetherian partial skew groupoid rings### Nystedt, Patrik

### Öinert, Johan

### Pinedo, Héctor

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 503, p. 433-452Article in journal (Refereed) Published
##### Abstract [en]

##### 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)
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##### Note

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.

Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.

Universidad Industrial de Santander, Escuela de Matemáticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia.

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.

Available online 14 February 2018

Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2019-05-28Bibliographically approvedOpen this publication in new window or tab >>Epsilon-strongly graded rings, separability and semisimplicity### Nystedt, Patrik

### Öinert, Johan

### Pinedo, Héctor

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 514, p. 1-24Article in journal (Refereed) Published
##### Abstract [en]

##### 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)
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Available from: 2018-10-29 Created: 2018-10-29 Last updated: 2019-05-28Bibliographically approved

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.

Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, SE-37179, Sweden.

Universidad Industrial de Santander, Escuela de Matemáticas, Carrera 27 Calle 9, Edificio Camilo Torres Apartado de correos 678, Bucaramanga, Colombia.

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 NÇstÇsescu, 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 SimÃ³n 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.

Open this publication in new window or tab >>Non-associative Ore extensions### Nystedt, Patrik

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.### Öinert, Johan

### Richter, Johan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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##### Note

Blekinge Institute of Technology, Department of Mathematics and Natural Sciences, Karlskrona, Sweden.

Mälardalen University, Academy of Education, Culture and Communication,Västerås, Sweden.

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.

First Online: 06 March 2018

Available from: 2018-06-15 Created: 2018-06-15 Last updated: 2019-05-28Bibliographically approvedOpen this publication in new window or tab >>Partial category actions on sets and topological spaces### Nystedt, Patrik

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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Available from: 2019-02-06 Created: 2019-02-06 Last updated: 2019-05-28Bibliographically approved

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.

Open this publication in new window or tab >>A proof of the law of sines using the law of cosines### Nystedt, Patrik

University West, Department of Engineering Science, Division of Mathematics, Computer and Surveying Engineering.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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Available from: 2017-12-12 Created: 2017-12-12 Last updated: 2019-05-23Bibliographically approved

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

Open this publication in new window or tab >>Fuzzy crossed product algebras### Nystedt, Patrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 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]

##### Keywords

Fuzzy ring, Fuzzy group, Cohomolgy
##### National Category

Algebra and Logic
##### Research subject

ENGINEERING, Mathematics
##### Identifiers

urn:nbn:se:hv:diva-8911 (URN)
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Available from: 2016-01-21 Created: 2016-01-21 Last updated: 2019-05-13Bibliographically approved

University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.

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.

Open this publication in new window or tab >>Outer Partial Actions and Partial Skew Group Rings### Nystedt, Patrik

### Öinert, Johan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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Available from: 2015-08-27 Created: 2015-08-27 Last updated: 2019-05-14Bibliographically approved

University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.

Lund University, Centre for Mathematical Sciences.

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.

Open this publication in new window or tab >>Simple semigroup graded rings### Nystedt, Patrik

### Oinert, Johan

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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Available from: 2015-05-30 Created: 2015-05-30 Last updated: 2017-12-04Bibliographically approved

University West, Department of Engineering Science, Division of Mechanical Engineering and Natural Sciences.

Lund Univ, Ctr Math Sci, SE-22100 Lund, Sweden.

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.

Open this publication in new window or tab >>A Proof of the Cosine Addition Formula Using the Law of Cosines### Nystedt, Patrik

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
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Available from: 2014-05-08 Created: 2014-05-08 Last updated: 2019-05-07Bibliographically approved

University West, Department of Engineering Science, Divison of Natural Sciences, Surveying and Mechanical Engineering.

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