Journal articles on the topic 'Rings and Algebras (math.RA)'
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Guo, Junying, and Xiaojiang Guo. "Algebras of right ample semigroups." Open Mathematics 16, no. 1 (August 3, 2018): 842–61. http://dx.doi.org/10.1515/math-2018-0075.
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AbstractStrict RA semigroups are common generalizations of ample semigroups and inverse semigroups. The aim of this paper is to study algebras of strict RA semigroups. It is proved that any algebra of strict RA semigroups with finite idempotents has a generalized matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. In particular, it is proved that any algebra of finite right ample semigroups has a generalized upper triangular matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. As its application, we determine when an algebra of strict RA semigroups (right ample monoids) is semiprimitive. Moreover, we prove that an algebra of strict RA semigroups (right ample monoids) is left self-injective iff it is right self-injective, iff it is Frobenius, and iff the semigroup is a finite inverse semigroup.
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CARVALHO, PAULA A. A. B., and IAN M. MUSSON. "MONOLITHIC MODULES OVER NOETHERIAN RINGS." Glasgow Mathematical Journal 53, no. 3 (August 1, 2011): 683–92. http://dx.doi.org/10.1017/s0017089511000267.
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AbstractWe study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.
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COYETTE, CÉCILE. "MAL'CEV–NEUMANN RINGS AND NONCROSSED PRODUCT DIVISION ALGEBRAS." Journal of Algebra and Its Applications 11, no. 03 (May 24, 2012): 1250052. http://dx.doi.org/10.1142/s0219498811005804.
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The first section of this paper yields a sufficient condition for a Mal'cev–Neumann ring of formal series to be a noncrossed product division algebra. This result is used in Sec. 2 to give an elementary proof of the existence of noncrossed product division algebras (of degree 8 or degree p2 for p any odd prime). The arguments are based on those of Hanke in [A direct approach to noncrossed product division algebras, thesis dissertation, Postdam (2001), An explicit example of a noncrossed product division algebra, Math. Nachr.251 (2004) 51–68, A twisted Laurent series ring that is a noncrossed product, Israel. J. Math.150 (2005) 199–2003].
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MASUOKA, AKIRA, and MAKOTO YANAGAWA. "×R-BIALGEBRAS ASSOCIATED WITH ITERATIVE q-DIFFERENCE RINGS." International Journal of Mathematics 24, no. 04 (April 2013): 1350030. http://dx.doi.org/10.1142/s0129167x13500304.
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Realizing the possibility suggested by Hardouin [Iterative q-difference Galois theory, J. Reine Angew. Math.644 (2010) 101–144], we show that her own Picard–Vessiot (PV) theory for iterative q-difference rings is covered by the (consequently, more general) framework, settled by Amano and Masuoka [Picard–Vessiot extensions of artinian simple module algebras, J. Algebra285 (2005) 743–767], of artinian simple module algebras over a cocommutative pointed Hopf algebra. An essential point is to represent iterative q-difference modules over an iterative q-difference ring R, by modules over a certain cocommutative ×R-bialgebra. Recall that the notion of ×R-bialgebras was defined by Sweedler [Groups of simple algebras, Publ. Math. Inst. Hautes Études Sci.44 (1974) 79–189], as a generalization of bialgebras.
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PAN, SHENGYONG, ZHEN PENG, and JIE ZHANG. "DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES." Glasgow Mathematical Journal 61, no. 03 (September 12, 2018): 557–73. http://dx.doi.org/10.1017/s0017089518000368.
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AbstractIn this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.
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Hermida-Alonso, José Ángel, Miguel V. Carriegos, Andrés Sáez-Schwedt, and Tomás Sánchez-Giralda. "On the regulator problem for linear systems over rings and algebras." Open Mathematics 19, no. 1 (January 1, 2021): 101–10. http://dx.doi.org/10.1515/math-2021-0002.
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Abstract The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).
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Ánh, Phạm Ngọc. "Skew Polynomial Rings: the Schreier Technique." Acta Mathematica Vietnamica 47, no. 1 (January 22, 2022): 5–17. http://dx.doi.org/10.1007/s40306-021-00466-7.
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AbstractSchreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc.145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras.
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Izelgue, L., and O. Ouzzaouit. "Hilbert rings and G(oldman)-rings issued from amalgamated algebras." Journal of Algebra and Its Applications 17, no. 02 (January 23, 2018): 1850023. http://dx.doi.org/10.1142/s0219498818500238.
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Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.
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Su, Dong, and Shilin Yang. "Automorphism groups of representation rings of the weak Sweedler Hopf algebras." AIMS Mathematics 7, no. 2 (2022): 2318–30. http://dx.doi.org/10.3934/math.2022131.
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<abstract><p>Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.</p></abstract>
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Guo, Junying, and Xiaojiang Guo. "Generalized Munn rings." Open Mathematics 20, no. 1 (January 1, 2022): 1066–81. http://dx.doi.org/10.1515/math-2022-0487.
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Abstract Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring R R , R R is primitive (semiprimitive and semiprime, respectively) if and only if so is M n ( R ) {M}_{n}\left(R) . As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.
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Fujita, Ryo, David Hernandez, Se-jin Oh, and Hironori Oya. "Isomorphisms among quantum Grothendieck rings and propagation of positivity." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 785 (February 15, 2022): 117–85. http://dx.doi.org/10.1515/crelle-2021-0088.
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Abstract Let ( 𝔤 , 𝗀 ) {\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀 {\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} and 𝒞 𝗀 {\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of 𝔤 {\mathfrak{g}} and 𝗀 {\mathsf{g}} , respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced 𝔤 {\mathfrak{g}} . In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q , t q,t -characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced 𝔤 {\mathfrak{g}} , and for any simple finite-dimensional modules in 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for 𝔤 {\mathfrak{g}} of type B n {\mathrm{B}_{n}} . In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all 𝔤 {\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀 {\mathsf{g}} and subalgebras of the quantum Grothendieck ring of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} .
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Guo, Junying, and Xiaojiang Guo. "Semiprimeness of semigroup algebras." Open Mathematics 19, no. 1 (January 1, 2021): 803–32. http://dx.doi.org/10.1515/math-2021-0026.
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Abstract Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.
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Fonseca, Graziela, Eneilson Fontes, and Grasiela Martini. "Multiplier Hopf algebras: Globalization for partial actions." International Journal of Algebra and Computation 30, no. 03 (November 26, 2019): 539–65. http://dx.doi.org/10.1142/s0218196720500101.
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In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.
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SNASHALL, NICOLE, and RACHEL TAILLEFER. "THE HOCHSCHILD COHOMOLOGY RING OF A CLASS OF SPECIAL BISERIAL ALGEBRAS." Journal of Algebra and Its Applications 09, no. 01 (February 2010): 73–122. http://dx.doi.org/10.1142/s0219498810003781.
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We consider a class of self-injective special biserial algebras ΛN over a field K and show that the Hochschild cohomology ring of dΛN is a finitely generated K-algebra. Moreover, the Hochschild cohomology ring of ΛN modulo nilpotence is a finitely generated commutative K-algebra of Krull dimension two. As a consequence the conjecture of [N. Snashall and Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004) 705–732], concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.
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Berest, Yuri, and Oleg Chalykh. "Quasi-invariants of complex reflection groups." Compositio Mathematica 147, no. 3 (September 27, 2010): 965–1002. http://dx.doi.org/10.1112/s0010437x10005063.
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AbstractWe introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.
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PATTANAYAK, S. K. "ON SOME STANDARD GRADED ALGEBRAS IN MODULAR INVARIANT THEORY." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350080. http://dx.doi.org/10.1142/s0219498813500801.
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For a finite-dimensional representation V of a finite group G over a field K we denote the graded algebra R ≔ ⨁d≥0 Rd; where Rd ≔ ( Sym d∣G∣V*)G. We study the standardness of R for the representations [Formula: see text], [Formula: see text], and [Formula: see text], where Vn denote the n-dimensional indecomposable representation of the cyclic group Cp over the Galois field 𝔽p, for a prime p. We also prove the standardness for the defining representation of all finite linear groups with polynomial rings of invariants. This is motivated by a question of projective normality raised in [S. S. Kannan, S. K. Pattanayak and P. Sardar, Projective normality of finite groups quotients, Proc. Amer. Math. Soc.137(3) (2009) 863–867].
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Bouchiba, Samir. "Local dimension theory of tensor products of algebras over a ring." Journal of Algebra and Its Applications 17, no. 06 (May 23, 2018): 1850106. http://dx.doi.org/10.1142/s0219498818501062.
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Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring [Formula: see text]. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field [Formula: see text] into the general setting of algebras over an arbitrary ring [Formula: see text]. For this sake, we introduce and study the notion of a fibered AF-ring over a ring [Formula: see text]. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings [Formula: see text] consisting of two [Formula: see text]-algebras [Formula: see text] and [Formula: see text] such that [Formula: see text], we introduce the inherent notion to [Formula: see text] of a [Formula: see text]-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product [Formula: see text]. As an application, we provide a formula for the Krull dimension of [Formula: see text] when either [Formula: see text] or [Formula: see text] is zero-dimensional as well as for the Krull dimension of [Formula: see text] when [Formula: see text] is a fibered AF-ring over the ring of integers [Formula: see text] with nonzero characteristic and [Formula: see text] is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of [Formula: see text] when [Formula: see text] is a Boolean ring. Actually, we prove that if [Formula: see text] and [Formula: see text] are rings such that [Formula: see text] is not trivial and [Formula: see text] is a Boolean ring, then dim[Formula: see text].
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LI, LIPING. "HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS." Glasgow Mathematical Journal 59, no. 2 (June 10, 2016): 401–20. http://dx.doi.org/10.1017/s0017089516000240.
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AbstractIn this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).
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Toumi, Mohamed Ali. "When the image of a derivation on a uniformly complete 𝑓-algebra is contained in the radical." Forum Mathematicum 32, no. 6 (November 1, 2020): 1561–73. http://dx.doi.org/10.1515/forum-2016-0082.
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AbstractIn 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.
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HABIBI, M., and A. MOUSSAVI. "NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS." Journal of Algebra and Its Applications 11, no. 04 (July 31, 2012): 1250080. http://dx.doi.org/10.1142/s0219498812500806.
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Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128–3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc.7 (1956) 35–48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x-1] are nil-Armendariz relative to any u.p.-monoid M, and we have nil (R[M]) = nil (R)[M].
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Paykan, Kamal, and Ahmad Moussavi. "Nilpotent elements and nil-Armendariz property of skew generalized power series rings." Asian-European Journal of Mathematics 10, no. 02 (August 2, 2016): 1750034. http://dx.doi.org/10.1142/s1793557117500346.
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Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid, and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we introduce and study the [Formula: see text]-nil-Armendariz condition on [Formula: see text], a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-nil-Armendariz. The [Formula: see text]-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring [Formula: see text] over a nil ring [Formula: see text] is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35–48]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.
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Klopsch, Benjamin, and Anitha Thillaisundaram. "Maximal Subgroups and Irreducible Representations of Generalized Multi-Edge Spinal Groups." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (April 17, 2018): 673–703. http://dx.doi.org/10.1017/s0013091517000451.
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AbstractLet p ≥ 3 be a prime. A generalized multi-edge spinal group $$G = \langle \{ a\} \cup \{ b_i^{(j)} {\rm \mid }1 \le j \le p,\, 1 \le i \le r_j\} \rangle \le {\rm Aut}(T)$$ is a subgroup of the automorphism group of a regular p-adic rooted tree T that is generated by one rooted automorphism a and p families $b^{(j)}_{1}, \ldots, b^{(j)}_{r_{j}}$ of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalizes the concepts of multi-edge spinal groups, including the widely studied GGS groups (named after Grigorchuk, Gupta and Sidki), and extended Gupta–Sidki groups that were introduced by Pervova [‘Profinite completions of some groups acting on trees, J. Algebra310 (2007), 858–879’]. Extending techniques that were developed in these more special cases, we prove: generalized multi-edge spinal groups that are torsion have no maximal subgroups of infinite index. Furthermore, we use tree enveloping algebras, which were introduced by Sidki [‘A primitive ring associated to a Burnside 3-group, J. London Math. Soc.55 (1997), 55–64’] and Bartholdi [‘Branch rings, thinned rings, tree enveloping rings, Israel J. Math.154 (2006), 93–139’], to show that certain generalized multi-edge spinal groups admit faithful infinite-dimensional irreducible representations over the prime field ℤ/pℤ.
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Sharma, Ram Parkash, and Anu. "SEMIALGEBRAS AND THEIR ALGEBRAS OF DIFFERENCES WITH PARTIAL GROUP ACTIONS ON THEM." Asian-European Journal of Mathematics 06, no. 03 (September 2013): 1350038. http://dx.doi.org/10.1142/s1793557113500381.
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Let A be a semialgebra defined in [R. P. Sharma, Anu and N. Singh, Partial group actions on semialgebras, Asian European J. Math.5(4) (2012), Article ID:1250060, 20pp.] over an additively cancellative and commutative semiring K. In additively cancellative semirings, the subtractive ideals play an important role. If P is a subtractive and G-prime ideal of an additively cancellative and yoked semiring A, where G is a finite group acting on A, then A has finitely many n(≤ |G|) minimal primes over P (see [R. P. Sharma and T. R. Sharma, G-prime ideals in semirings and their skew group rings, Comm. Algebra34 (2006) 4459–4465], Lemma 3.6, a result analogous to Lemma 3.2 of Passman [D. S. Passman, It's essentially Maschke's theorem, Rocky Mountain J. Math.13 (1983) 37–54]). Consider a subtractive partial action α of a finite group G on A such that each Dg is generated by a central idempotent 1g of A and the intersection D = ⋂g∈G,Dg≠0Dg of nonzero Dg's is nonzero. It is not necessary that number of minimal primes in Spec A over a subtractive and α-prime ideal P of a yoked semiring A is less than or equal to the order of the group, if 1d ∈ P (Example 3.2). However, we show that the result is true if 1d ∉ P (Corollary 3.1). We also study the prime ideals of the partial fixed subsemiring Aα of A.
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Iyama, Osamu, and Michael Wemyss. "Reduction of triangulated categories and maximal modification algebras for cAn singularities." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 738 (May 1, 2018): 149–202. http://dx.doi.org/10.1515/crelle-2015-0031.
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Abstract In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If \mathcal{C} is such a category, we say that \mathcal{C} is Calabi–Yau with \dim\mathcal{C}\leq 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k=\mathbb{C} we obtain many autoequivalences of the derived category of the \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R .
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MEINEL, JOANNA, and CATHARINA STROPPEL. "GOLDIE RANK OF PRIMITIVE QUOTIENTS VIA LATTICE POINT ENUMERATION." Glasgow Mathematical Journal 55, A (October 2013): 149–68. http://dx.doi.org/10.1017/s0017089513000566.
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AbstractLet k be an algebraically closed field of characteristic zero. I. M. Musson and M. Van den Bergh (Mem. Amer. Math. Soc., vol. 136, 1998, p. 650) classify primitive ideals for rings of torus invariant differential operators. This classification applies in particular to subquotients of localized extended Weyl algebras $\mathcal{A}_{r,n-r}=k[x_1,\ldots,x_r,x_{r+1}^{\pm1}, \ldots, x_{n}^{\pm1},\partial_1,\ldots,\partial_n],$ where it can be made explicit in terms of convex geometry. We recall these results and then turn to the corresponding primitive quotients and study their Goldie ranks. We prove that the primitive quotients fall into finitely many families whose Goldie ranks are given by a common quasi-polynomial and then realize these quasi-polynomials as Ehrhart quasi-polynomials arising from convex geometry.
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Herrmann, Christian, Yasuyuki Tsukamoto, and Martin Ziegler. "On the consistency problem for modular lattices and related structures." International Journal of Algebra and Computation 26, no. 08 (December 2016): 1573–95. http://dx.doi.org/10.1142/s0218196716500697.
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The consistency problem for a class of algebraic structures asks for an algorithm to decide, for any given conjunction of equations, whether it admits a non-trivial satisfying assignment within some member of the class. For the variety of all groups, this is the complement of the triviality problem, shown undecidable by by Adyan [Algorithmic unsolvability of problems of recognition of certain properties of groups. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 533–535] and Rabin [Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958) 172–194]. For the class of finite groups, it amounts to the triviality problem for profinite completions, shown undecidable by Bridson and Wilton [The triviality problem for profinite completions, Invent. Math. 202 (2015) 839–874]. We derive unsolvability of the consistency problem for the class of (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite-dimensional vector spaces over a fixed or arbitrary field of characteristic [Formula: see text] and expansions thereof, e.g. the class of subspace ortholattices of finite-dimensional Hilbert spaces. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings with unit and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann–Cayley algebra and to functional and embedded multivalued dependencies in databases.
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Toumi, Mohamed Ali. "CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS." Asian-European Journal of Mathematics 05, no. 03 (September 2012): 1250045. http://dx.doi.org/10.1142/s1793557112500453.
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Let A be an ℓ-algebra and let θ and ϕ be two endomorphisms of A. The couple (θ, ϕ) is called to be separating if xy = 0 implies θ(x)ϕ(y) = 0. If in addition θ and ϕ are ring endomorphisms of A, then the couple (θ, ϕ) is said to be ring-separating. An additive mapping δ : A → A is called (θ, ϕ)-separating derivation on A if there exists a (θ, ϕ)-separating couple with δ(xy) = δ(x)θ(y) + ϕ(x)δ(y), holds for all x, y ∈ A. If an addition θ, ϕ and δ are continuous, then δ is called a continuous (θ, ϕ)-ring-separating derivation. If in addition the couple (θ, ϕ) is ring-separating then δ is called a continuous (θ, ϕ)-ring-separating derivation. An additive mapping F : A → A is called a continuous generalized (θ, ϕ)-separating derivation on A if F is continuous mapping and if there exists a derivation d : A → A such that θ and ϕ are continuous, (θ, ϕ) is a separating couple and F(xy) = F(x)θ(y) + ϕ(x)d(y), holds for all x, y ∈ A. In this paper, we give a description of continuous (θ, ϕ)-ring-separating derivations on some ℓ-algebras. This generalizes a well-known theorem by Colville, Davis, and Keimel [Positive derivations on f-rings, J. Austral. Math. Soc23 (1977) 371–375] and generalizes the results of Boulabiar in [Positive derivations on almost f-rings, Order19 (2002) 385–395], Ben Amor [On orthosymmetric bilinear maps, Positivity14(1) (2010) 123–130] and Toumi et al. in [Order bounded derivations on Archimedean almost f-algebras, Positivity14(2) (2010) 239–245]. Moreover, inspiring from [Toumi, Order-bounded generalized derivations on Archimedean almost f-algebras, Commun. Algebra38(1) (2010) 154–164], it is shown that the notion of continuous generalized (θ, ϕ)-separating derivation on an archimedean almost f-algebra A is the concept of generalized θ-multiplier, that is an additive mapping satisfying F(xyz) = F(x)θ(yz), for all x, y, z ∈ A. In the case where A is an archimedean f-algebra, the situation improves. Indeed, the collection of all continuous generalized (θ, ϕ)-separating derivation on A coincides with the concept of θ-multiplier, that is an additive mapping satisfying F(xy) = F(x)θ(y), for all x, y ∈ A. If in addition A is a Dedekind complete vector lattice and θ is a positive mapping, then the set of all order bounded generalized of the form (θ, ϕ)-separating derivations on A, under composition, is an archimedean lattice-ordered algebra.
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Zhao, Xiangui, and Yang Zhang. "A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings." Open Mathematics 13, no. 1 (May 6, 2015). http://dx.doi.org/10.1515/math-2015-0028.
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AbstractSignature-based algorithms are efficient algorithms for computing Gröbner-Shirshov bases in commutative polynomial rings, and some noncommutative rings. In this paper, we first define skew solvable polynomial rings, which are generalizations of solvable polynomial algebras and (skew) PBW extensions. Then we present a signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings over fields.
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Cui, Jian, and Peter Danchev. "On Strongly pi-Regular Rings with Involution." Communications in Mathematics Volume 31 (2023), Issue 1 (November 11, 2022). http://dx.doi.org/10.46298/cm.10273.
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Recall that a ring R is called strongly pi-regular if, for every a in R, there is a positive integer n, depending on a, such that a^n belongs to the intersection of a^{n+1}R and Ra^{n+1}. In this paper we give a further study of the notion of a strongly pi-star-regular ring, which is the star-version of strongly pi-regular rings and which was originally introduced by Cui-Wang in J. Korean Math. Soc. (2015). We also establish various properties of these rings and give several new characterizations in terms of (strong) pi-regularity and involution. Our results also considerably extend recent ones in the subject due to Cui-Yin in Algebra Colloq. (2018) proved for pi-star-regular rings and due to Cui-Danchev in J. Algebra Appl. (2020) proved for star-periodic rings.
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ABDİOĞLU, Cihat, Shakir ALI, and Mohammad Salahuddin KHAN. "On ideals of prime rings involving $n$-skew commuting additive mappings with applications." Hacettepe Journal of Mathematics and Statistics, October 1, 2022, 1237–47. http://dx.doi.org/10.15672/hujms.776236.
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Let $n > 1 $ be a fixed positive integer and $S$ be a subset of a ring $R$. A mapping $\zeta$ of a ring $R$ into itself is called $n$-skew-commuting on $S$ if $\zeta(x)x^{n} + x^{n}\zeta(x)=0$, $\forall$ $x\in S.$ The main aim of this paper is to describe $n$-skew-commuting mappings on appropriate subsets of $R$. With this, many known results can be either generalized or deduced. In particular, this solves the conjecture in [M. Nadeem, M. Aslam and M.A. Javed, On $2$-skew commuting additive mappings of prime rings, Gen. Math. Notes, 2015]. The second main result of this paper is concerned with a pair of linear mappings of $C^*$-algebras. We show that here, if $C^*$-Algebra admits a pair of linear mappings $f$ and $g$ such that $f(x)x^* + x^*g(x) \in Z(A)$ for all $x \in A,$ then both $f$ and $g$ must be zero. As the applications of first main result (Theorem $2.1$) and apart from proving some other results, we characterize the linear mappings on primitive $C^*$-algebras. Furthermore, we provide an example to show that the assumed restrictions cannot be relaxed.