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Journal articles on the topic 'Algebras- Commutative rings'

Author: Grafiati

Published: 6 September 2023

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1

Finkel, Olivier, and Stevo Todorčević. "A hierarchy of tree-automatic structures." Journal of Symbolic Logic 77, no. 1 (March 2012): 350–68. http://dx.doi.org/10.2178/jsl/1327068708.

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AbstractWe consider ωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωn for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω2-automatic (resp. ωn-automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ωn-automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many ωn-automatic, hence also ω-tree-automatic, atomless boolean algebras , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].

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2

Flaut, Cristina, and Dana Piciu. "Some Examples of BL-Algebras Using Commutative Rings." Mathematics 10, no. 24 (December 13, 2022): 4739. http://dx.doi.org/10.3390/math10244739.

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BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2r+1 elements, r≥2, and BL-chains with k elements, k≥4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size (2≤n≤5).

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3

Tuganbaev, A. A. "Quaternion algebras over commutative rings." Mathematical Notes 53, no. 2 (February 1993): 204–7. http://dx.doi.org/10.1007/bf01208328.

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4

Tambour, Torbjörn. "S-algebras and commutative rings." Journal of Pure and Applied Algebra 82, no. 3 (October 1992): 289–313. http://dx.doi.org/10.1016/0022-4049(92)90173-d.

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5

Zhou, Chaoyuan. "Acyclic Complexes and Graded Algebras." Mathematics 11, no. 14 (July 19, 2023): 3167. http://dx.doi.org/10.3390/math11143167.

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We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative N-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra.

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6

CHAKRABORTY, S., R. V. GURJAR, and M. MIYANISHI. "PURE SUBRINGS OF COMMUTATIVE RINGS." Nagoya Mathematical Journal 221, no. 1 (March 2016): 33–68. http://dx.doi.org/10.1017/nmj.2016.2.

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7

Macoosh, R., and R. Raphael. "Totally Integrally Closed Azumaya Algebras." Canadian Mathematical Bulletin 33, no. 4 (December 1, 1990): 398–403. http://dx.doi.org/10.4153/cmb-1990-065-5.

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AbstractEnochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.

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8

Cimprič, Jakob. "A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings." Canadian Mathematical Bulletin 52, no. 1 (March 1, 2009): 39–52. http://dx.doi.org/10.4153/cmb-2009-005-4.

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AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

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9

Bix, Robert. "Separable alternative algebras over commutative rings." Journal of Algebra 92, no. 1 (January 1985): 81–103. http://dx.doi.org/10.1016/0021-8693(85)90146-2.

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10

Scedrov, Andre, and Philip Scowcroft. "Decompositions of finitely generated modules over C(X): sheaf semantics and a decision procedure." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (March 1988): 257–68. http://dx.doi.org/10.1017/s0305004100064823.

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In the theory of operator algebras the rings of finite matrices over such algebras play a very important role (see [10]). For commutative operator algebras, the Gelfand-Naimark representation allows one to concentrate on matrices over rings of continuous complex functions on compact Hausdorif spaces.

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11

El Khalfi, Abdelhaq, and Najib Mahdou. "Almost Bézout rings and almost GCD-rings." Asian-European Journal of Mathematics 13, no. 06 (April 1, 2019): 2050107. http://dx.doi.org/10.1142/s1793557120501077.

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In this paper, we study the possible transfer of the property of being an [Formula: see text]-ring to trivial ring extensions and amalgamated algebras along an ideal. Also, we extend the notion of an almost GCD-domain to the context of arbitrary rings, and we study the possible transfer of this notion to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

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12

Kosters, Michiel. "Algebras with only finitely many subalgebras." Journal of Algebra and Its Applications 14, no. 06 (April 21, 2015): 1550086. http://dx.doi.org/10.1142/s0219498815500863.

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Let R be a commutative ring. A not necessarily commutative R-algebra A is called futile if it has only finitely many R-subalgebras. In this paper, we relate the notion of futility to familiar properties of rings and modules. We do this by first reducing to the case where A is commutative. Then we refine the description of commutative futile algebras from Dobbs, Picavet and Picavet-L'Hermite.

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13

Farkas, Daniel R., C. D. Feustel, and Edward L. Green. "Synergy in the Theories of Gröbner Bases and Path Algebras." Canadian Journal of Mathematics 45, no. 4 (August 1, 1993): 727–39. http://dx.doi.org/10.4153/cjm-1993-041-8.

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14

Lezama, Oswaldo, Juan Pablo Acosta, Cristian Chaparro, Ingrid Ojeda, and César Venegas. "ORE AND GOLDIE THEOREMS FOR SKEW PBW EXTENSIONS." Asian-European Journal of Mathematics 06, no. 04 (December 2013): 1350061. http://dx.doi.org/10.1142/s1793557113500617.

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Many rings and algebras arising in quantum mechanics can be interpreted as skew Poincaré–Birkhoff–Witt (PBW) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper, we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand–Kirillov conjecture for the skew quantum polynomials.

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15

Abuhlail, Jawad Y., José Gómez-Torrecillas, and Robert Wisbauer. "Dual coalgebras of algebras over commutative rings." Journal of Pure and Applied Algebra 153, no. 2 (October 2000): 107–20. http://dx.doi.org/10.1016/s0022-4049(99)00088-2.

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16

Loos, Ottmar. "Generically algebraic Jordan algebras over commutative rings." Journal of Algebra 297, no. 2 (March 2006): 474–529. http://dx.doi.org/10.1016/j.jalgebra.2005.11.021.

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17

Nordstrom, Hans, and Jennifer A. Firkins Nordstrom. "Leavitt path algebras over arbitrary unital rings and algebras." Journal of Algebra and Its Applications 19, no. 06 (May 31, 2019): 2050107. http://dx.doi.org/10.1142/s0219498820501078.

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We expand the work of Tomforde by further extending the construction of Leavitt path algebras (LPAs) over arbitrary associative, unital rings. We show that many of the results over a commutative ring hold in the more general setting, provide some useful generalizations of prior results, and give a definition for an iterated Leavitt path extension in our context.

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18

STEINBERG, BENJAMIN. "CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS." Journal of the Australian Mathematical Society 104, no. 3 (March 28, 2018): 403–11. http://dx.doi.org/10.1017/s1446788717000374.

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The author has previously associated to each commutative ring with unit$R$and étale groupoid$\mathscr{G}$with locally compact, Hausdorff and totally disconnected unit space an$R$-algebra$R\,\mathscr{G}$. In this paper we characterize when$R\,\mathscr{G}$is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

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19

Jin, Hailan, Tai Keun Kwak, Yang Lee, and Zhelin Piao. "A Property Satisfying Reducedness over Centers." Algebra Colloquium 28, no. 03 (July 26, 2021): 453–68. http://dx.doi.org/10.1142/s1005386721000353.

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This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

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20

Nikolopoulos, Christos, and Panagiotis Nikolopoulos. "Generalizations of the primitive element theorem." International Journal of Mathematics and Mathematical Sciences 14, no. 3 (1991): 463–70. http://dx.doi.org/10.1155/s0161171291000637.

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In this paper we generalize the primitive element theorem to the generation of separable algebras over fields and rings. We prove that any finitely generated separable algebra over an infinite field is generated by two elements and if the algebra is commutative it can be generated by one element. We then derive similar results for finitely generated separable algebras over semilocal rings.

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21

Parmenter, M. M., E. Spiegel, and P. N. Stewart. "The Periodic Radical of Group Rings and Incidence Algebras." Canadian Mathematical Bulletin 41, no. 4 (December 1, 1998): 481–87. http://dx.doi.org/10.4153/cmb-1998-063-4.

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AbstractLet R be a ring with 1 and P(R) the periodic radical of R. We obtain necessary and sufficient conditions for P(RG) = 0 when RG is the group ring of an FC group G and R is commutative. We also obtain a complete description of when I(X, R) is the incidence algebra of a locally finite partially ordered set X and R is commutative.

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22

Nagayama, Misao. "On Boolean algebras and integrally closed commutative regular rings." Journal of Symbolic Logic 57, no. 4 (December 1992): 1305–18. http://dx.doi.org/10.2307/2275369.

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AbstractIn this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai, partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S′(ai, l) for l < n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ϵ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.

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23

MESYAN, ZACHARY, and LIA VAŠ. "TRACES ON SEMIGROUP RINGS AND LEAVITT PATH ALGEBRAS." Glasgow Mathematical Journal 58, no. 1 (July 21, 2015): 97–118. http://dx.doi.org/10.1017/s0017089515000087.

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AbstractThe trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotients of contracted semigroup rings), where we describe all linear traces in terms of central maps on graph inverse semigroups and, under mild assumptions, those Leavitt path algebras that admit faithful traces.

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24

ROBINSON, ALAN, and SARAH WHITEHOUSE. "Operads and Γ-homology of commutative rings." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 2 (March 2002): 197–234. http://dx.doi.org/10.1017/s0305004102005534.

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We introduce Γ-homology, the natural homology theory for E∞-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from André–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

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25

Ludkowski, Sergey Victor. "Nonassociative Algebras, Rings and Modules over Them." Mathematics 11, no. 7 (April 3, 2023): 1714. http://dx.doi.org/10.3390/math11071714.

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The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials, commutative nonassociative algebras and their modules, nonassociative cyclic algebras, rings obtained as nonassociative cyclic extensions, nonassociative Ore extensions of hom-associative algebras and modules over them, and von Neumann finiteness for nonassociative algebras. Furthermore, there are outlined nonassociative algebras and rings and modules over them related to harmonic analysis on nonlocally compact groups, nonassociative algebras with conjugation, representations and closures of nonassociative algebras, and nonassociative algebras and modules over them with metagroup relations. Moreover, classes of Akivis, Sabinin, Malcev, Bol, generalized Cayley–Dickson, and Zinbiel-type algebras are provided. Sources also are reviewed on near to associative nonassociative algebras and modules over them. Then, there are the considered applications of nonassociative algebras and modules over them in cryptography and coding, and applications of modules over nonassociative algebras in geometry and physics. Their interactions are discussed with more classical nonassociative algebras, such as of the Lie, Jordan, Hurwitz and alternative types.

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26

Loos, Ottmar, Holger Petersson, and Michel Racine. "Inner derivations of alternative algebras over commutative rings." Algebra & Number Theory 2, no. 8 (November 23, 2008): 927–68. http://dx.doi.org/10.2140/ant.2008.2.927.

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27

Ivanov, Sergei O., Fedor Pavutnitskiy, Vladislav Romanovskii, and Anatolii Zaikovskii. "On homology of Lie algebras over commutative rings." Journal of Algebra 586 (November 2021): 99–139. http://dx.doi.org/10.1016/j.jalgebra.2021.06.019.

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28

Benson, Dave, Srikanth B. Iyengar, and Henning Krause. "Module categories for group algebras over commutative rings." Journal of K-Theory 11, no. 2 (March 6, 2013): 297–329. http://dx.doi.org/10.1017/is013001031jkt214.

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AbstractWe develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.

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29

Aviñö Diaz, Maria Alicia, and Raymundo Bautista Ramos. "On artin algebras over artinian commutative local rings." Archiv der Mathematik 66, no. 5 (May 1996): 366–71. http://dx.doi.org/10.1007/bf01781554.

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30

Stokes, Timothy. "Conjugate polynomials over quadratic algebras." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 52, no. 2 (April 1992): 154–74. http://dx.doi.org/10.1017/s1446788700034327.

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AbstractThis paper gives variants of results from classical algebraic geometry and commutative algebra for quadratic algebras with conjugation. Quadratic algebras are essentially two-dimensional algebras with identity over commutative rings with identity, on which a natural operation of conjugation may be defined. We define the ring of conjugate polynomials over a quadratic algebra, and define c-varieties. In certain cases a close correspondence between standard varieties and c-varieties is demonstrated, and we establish a correspondence between conjugate and standard polynomials, which leads to variants of the Hilbert Nullstellensatz if the commutativering is an algebraically closed field. These results may be applied to automated Euclidean geometry theorem proving.

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31

Pavlov, Dmitri, and Jakob Scholbach. "SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA." Journal of the Institute of Mathematics of Jussieu 18, no. 4 (May 25, 2018): 707–58. http://dx.doi.org/10.1017/s1474748017000202.

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This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

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32

Huang, Hongdi, Yuanlin Li, and Gaohua Tang. "On *-clean non-commutative group rings." Journal of Algebra and Its Applications 15, no. 08 (July 24, 2016): 1650150. http://dx.doi.org/10.1142/s0219498816501504.

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A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

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33

Sanghare, Mamadou. "Subrings of I-rings and S-rings." International Journal of Mathematics and Mathematical Sciences 20, no. 4 (1997): 825–27. http://dx.doi.org/10.1155/s0161171297001130.

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LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=K.[000100000]+K.[000000100]is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ringRis of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.

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34

MESNAGER, SIHEM. "TEST OF EPIMORPHISM FOR FINITELY GENERATED MORPHISMS BETWEEN AFFINE ALGEBRAS OVER COMPUTATIONAL RINGS." Journal of Algebra and Its Applications 04, no. 04 (August 2005): 405–19. http://dx.doi.org/10.1142/s0219498805001253.

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In this paper, based on a characterization of epimorphisms of R-algebras given by Roby [15], we bring an algorithm testing whether a given finitely generated morphism f : A → B, where A and B are finitely presented affine algebras over the same Nœtherian commutative ring R, is an epimorphism of R-algebras or not. We require two computational conditions on R, which we call a computational ring.

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35

Futorny, Vyacheslav, and João Schwarz. "Holonomic modules for rings of invariant differential operators." International Journal of Algebra and Computation 31, no. 04 (April 10, 2021): 605–22. http://dx.doi.org/10.1142/s0218196721500296.

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We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

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36

GHAHRAMANI, HOGER. "ON RINGS DETERMINED BY ZERO PRODUCTS." Journal of Algebra and Its Applications 12, no. 08 (July 31, 2013): 1350058. http://dx.doi.org/10.1142/s0219498813500588.

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Let [Formula: see text] be a ring. We say that [Formula: see text] is zero product determined if for every additive group [Formula: see text] and every bi-additive map [Formula: see text] the following holds: if ϕ(a, b) = 0 whenever ab = 0, then there exists an additive map [Formula: see text] such that ϕ(a, b) = T(ab) for all [Formula: see text]. In this paper, first we study the properties of zero product determined rings and show that semi-commutative and non-commutative rings are not zero product determined. Then, we will examine whether the rings with a nontrivial idempotent are zero product determined. As applications of the above results, we prove that simple rings with a nontrivial idempotent, full matrix rings and some classes of operator algebras are zero product determined rings and discuss whether triangular rings and upper triangular matrix rings are zero product determined.

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CAO, YOU'AN, DEZHI JIANG, and JUNYING WANG. "AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS." International Journal of Algebra and Computation 17, no. 03 (May 2007): 527–55. http://dx.doi.org/10.1142/s021819670700372x.

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Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

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38

Zhou, Jinming, Dengyin Wang, and Qinghua Zhang. "Square-zero derivations of matrix algebras over commutative rings." Linear and Multilinear Algebra 58, no. 2 (February 2010): 239–43. http://dx.doi.org/10.1080/03081080802443091.

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Cárdenas, Raggi A., and L. Salmerón. "Quivers for algebras over commutative noetherian complete local rings." Communications in Algebra 15, no. 12 (January 1987): 2617–23. http://dx.doi.org/10.1080/00927878708823554.

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Chen, Zhengxin, and Dengyin Wang. "Derivations of Certain Nilpotent Lie Algebras Over Commutative Rings." Communications in Algebra 39, no. 10 (October 2011): 3736–52. http://dx.doi.org/10.1080/00927872.2010.512581.

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Bezhanishvili, Guram, Vincenzo Marra, Patrick J. Morandi, and Bruce Olberding. "Idempotent generated algebras and Boolean powers of commutative rings." Algebra universalis 73, no. 2 (February 5, 2015): 183–204. http://dx.doi.org/10.1007/s00012-015-0321-8.

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42

Poroshenko, E. N. "Elementary Equivalence of Partially Commutative Lie Rings and Algebras." Algebra and Logic 56, no. 4 (September 2017): 348–52. http://dx.doi.org/10.1007/s10469-017-9455-4.

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Chalykh, O. A., and A. P. Veselov. "Commutative rings of partial differential operators and Lie algebras." Communications in Mathematical Physics 126, no. 3 (January 1990): 597–611. http://dx.doi.org/10.1007/bf02125702.

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Göbel, Rüdiger, and Warren May. "Four submodules suffice for realizing algebras over commutative rings." Journal of Pure and Applied Algebra 65, no. 1 (July 1990): 29–43. http://dx.doi.org/10.1016/0022-4049(90)90098-3.

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45

Enright, T. J. "Representations of Lie Algebras Defined over Commutative Rings I." Journal of Algebra 172, no. 3 (March 1995): 640–70. http://dx.doi.org/10.1006/jabr.1995.1064.

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46

Rossmanith, Richard. "Lie Centre-by-Metabelian Group Algebras over Commutative Rings." Journal of Algebra 251, no. 2 (May 2002): 503–8. http://dx.doi.org/10.1006/jabr.2001.9132.

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47

Ahmed, Khondokar M., SK Rasel, Jyoti Das, Saraban Tahura, and Salma Nasrin. "Study of Graded Algebras and General Linear Group with Lie Superalgebras and R-Algebra." Dhaka University Journal of Science 68, no. 1 (January 30, 2020): 1–5. http://dx.doi.org/10.3329/dujs.v68i1.54590.

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Some elements of theory of Z2 graded rings, modules and algebras. Z2-graded tensor algebra, Lie superalgrbras and matrices with entries in a Z2-graded commutative ring are treated in our present paper. At last a Theorem 4.4.on the set of square matrices in the graded R-algebra , MR-[m I n] is established. Dhaka Univ. J. Sci. 68(1): 1-5, 2020 (January)

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Birkenmeier, Gary F., Jae Keol Park, and S. Tariq Rizvi. "Hulls of Ring Extensions." Canadian Mathematical Bulletin 53, no. 4 (December 1, 2010): 587–601. http://dx.doi.org/10.4153/cmb-2010-065-9.

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AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

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Bremner, Murray, and Vladimir Dotsenko. "Distributive laws between the operads Lie and Com." International Journal of Algebra and Computation 30, no. 08 (August 20, 2020): 1565–76. http://dx.doi.org/10.1142/s021819672050054x.

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Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad.

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50

Nelis, Peter. "Schur and Projective Schur Groups of Number Rings." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 540–58. http://dx.doi.org/10.4153/cjm-1991-033-5.

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The Schur or projective Schur group of a field consists of the classes of central simple algebras which occur in the decomposition of a group algebra or a twisted group algebra. For number fields, the projective Schur group has been determined in [8], whereas the Schur group is extensively studied in [25]. Recently, some authors have generalized these concepts to commutative rings. One then studies the classes of Azumaya algebras which are epimorphic images of a group ring or a twisted group ring. Though several properties of the Schur or projective Schur group defined in this way have been obtained, they remain rather obscure objects.

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