Relevant bibliographies by topics / Commutative Rings and Algebras

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Commutative Rings and Algebras'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 August 2024

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Journal articles on the topic "Commutative Rings and Algebras"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Commutative Rings and Algebras"

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Malec, Sara. "Intersection Algebras and Pointed Rational Cones." Digital Archive @ GSU, 2013. http://digitalarchive.gsu.edu/math_diss/14.

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In this dissertation we study the algebraic properties of the intersection algebra of two ideals I and J in a Noetherian ring R. A major part of the dissertation is devoted to the finite generation of these algebras and developing methods of obtaining their generators when the algebra is finitely generated. We prove that the intersection algebra is a finitely generated R-algebra when R is a Unique Factorization Domain and the two ideals are principal, and use fans of cones to find the algebra generators. This is done in Chapter 2, which concludes with introducing a new class of algebras called fan algebras. Chapter 3 deals with the intersection algebra of principal monomial ideals in a polynomial ring, where the theory of semigroup rings and toric ideals can be used. A detailed investigation of the intersection algebra of the polynomial ring in one variable is obtained. The intersection algebra in this case is connected to semigroup rings associated to systems of linear diophantine equations with integer coefficients, introduced by Stanley. In Chapter 4, we present a method for obtaining the generators of the intersection algebra for arbitrary monomial ideals in the polynomial ring.
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Sekaran, Rajakrishnar. "Fuzzy ideals in commutative rings." Thesis, Rhodes University, 1995. http://hdl.handle.net/10962/d1005221.

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In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.
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Hasse, Erik Gregory. "Lowest terms in commutative rings." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6433.

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Putting fractions in lowest terms is a common problem for basic algebra courses, but it is rarely discussed in abstract algebra. In a 1990 paper, D.D. Anderson, D.F. Anderson, and M. Zafrullah published a paper called Factorization in Integral Domains, which summarized the results concerning different factorization properties in domains. In it, they defined an LT domain as one where every fraction is equal to a fraction in lowest terms. That is, for any x/y in the field of fractions of D, there is some a/b with x/y=a/b and the greatest common divisor of a and b is 1. In addition, R. Gilmer included a brief exercise concerning lowest terms over a domain in his book Multiplicative Ideal Theory. In this thesis, we expand upon those definitions. First, in Chapter 2 we make a distinction between putting a fraction in lowest terms and reducing it to lowest terms. In the first case, we simply require the existence of an equal fraction which is in lowest terms, while the second requires an element which divides both the numerator and the denominator to reach lowest terms. We also define essentially unique lowest terms, which requires a fraction to have only one lowest terms representation up to unit multiples. We prove that a reduced lowest terms domain is equivalent to a weak GCD domain, and that a domain which is both a reduced lowest terms domain and a unique lowest terms domain is equivalent to a GCD domain. We also provide an example showing that not every domain is a lowest terms domain as well as an example showing that putting a fraction in lowest terms is a strictly weaker condition than reducing it to lowest terms. Next, in Chapter 3 we discuss how lowest terms in a domain interacts with the polynomial ring. We prove that if D[T] is a unique lowest terms domain, then D must be a GCD domain. We also provide an alternative approach to some of the earlier results using the group of divisibility. So far, all fractions have been representatives of the field of fractions of a domain. However, in Chapter 4 we examine fractions in other localizations of a domain. We define a necessary and sufficient condition on the multiplicatively closed set, and then examine how this relates to existing properties of multiplicatively closed sets. Finally, in Chapter 5 we briefly examine lowest terms in rings with zero divisors. Because many properties of GCDs do not hold in such rings, this proved difficult. However, we were able to prove some results from Chapter 2 in this more general case.
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Bell, Kathleen. "Cayley Graphs of PSL(2) over Finite Commutative Rings." TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2102.

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Hadwiger's conjecture is one of the deepest open questions in graph theory, and Cayley graphs are an applicable and useful subtopic of algebra. Chapter 1 will introduce Hadwiger's conjecture and Cayley graphs, providing a summary of background information on those topics, and continuing by introducing our problem. Chapter 2 will provide necessary definitions. Chapter 3 will give a brief survey of background information and of the existing literature on Hadwiger's conjecture, Hamiltonicity, and the isoperimetric number; in this chapter we will explore what cases are already shown and what the most recent results are. Chapter 4 will give our decomposition theorem about PSL2 (R). Chapter 5 will continue with corollaries of the decomposition theorem, including showing that Hadwiger's conjecture holds for our Cayley graphs. Chapter 6 will finish with some interesting examples.
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Granger, Ginger Thibodeaux. "Properties of R-Modules." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc500710/.

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This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include existence theorems for R-modules and properties of submodules in R-modules. The third and final chapter presents an example which illustrates how a ring R, may be regarded as an R-module and speaks of the direct sum of ideals of a ring as a direct sum of submodules.
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Oman, Gregory Grant. "A generalization of Jónsson modules over commutative rings with identity." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1164331653.

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Coughlin, Heather. "Classes of normal monomial ideals /." view abstract or download file of text, 2004. http://wwwlib.umi.com/cr/uoregon/fullcit?p3147816//.

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Thesis (Ph. D.)--University of Oregon, 2004.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 85-86). Also available for download via the World Wide Web; free to University of Oregon users.
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Green, Ellen Yvonne. "Characterizing the strong two-generators of certain Noetherian domains." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1539.

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Salt, Brittney M. "MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/31.

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This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.
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Race, Denise T. (Denise Tatsch). "Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors." Thesis, North Texas State University, 1987. https://digital.library.unt.edu/ark:/67531/metadc331394/.

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This dissertation focuses on the significance of containment relations between the above mentioned classes of ideals. The main problem considered in Chapter II is determining conditions which lead a ring to be a P-ring, D-ring, or AM-ring when every regular ideal is a P-ideal, D-ideal, or AM-ideal, respectively. We also consider containment relations between classes of regular ideals which guarantee that the ring is a quasi-valuation ring. We continue this study into the third chapter; in particular, we look at the conditions in a quasi-valuation ring which lead to a = Jr, sr - f, and a = v. Furthermore we give necessary and sufficient conditions that a ring be a discrete rank one quasi-valuation ring. For example, if R is Noetherian, then ft = J if and only if R is a discrete rank one quasi-valuation ring.
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Books on the topic "Commutative Rings and Algebras"

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Gelʹfand, I. M. Commutative normed rings. Providence, RI: AMS Chelsea Publishing, 2003.

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Gelʹfand, I. M. Commutative normed rings. Providence, RI: AMS Chelsea Publishing, 2003.

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Gelʹfand, I. M. Commutative normed rings. Providence, RI: American Mathematical Society, 1999.

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Kaplansky, Irving. Commutative rings. Washington, N.J: Polygonal Pub. House, 1994.

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Bosch, Siegfried. Algebraic Geometry and Commutative Algebra. London: Springer London, 2013.

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service), SpringerLink (Online, ed. Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday. New York, NY: Springer New York, 2013.

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Kunz, Ernst. Introduction to Commutative Algebra and Algebraic Geometry. New York, NY: Springer New York, 2013.

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Glaz, Sarah. Commutative coherent rings. Berlin: Springer-Verlag, 1989.

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Lee, John. Commutative rings: New research. New York: Nova Science Publishers, 2009.

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service), SpringerLink (Online, ed. Categories and Commutative Algebra. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2011.

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Book chapters on the topic "Commutative Rings and Algebras"

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Cohn, P. M. "Commutative Rings." In Basic Algebra, 347–96. London: Springer London, 2003. http://dx.doi.org/10.1007/978-0-85729-428-9_10.

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Kempf, George R. "Commutative rings." In Algebraic Structures, 141–43. Wiesbaden: Vieweg+Teubner Verlag, 1995. http://dx.doi.org/10.1007/978-3-322-80278-1_18.

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Kadison, Lars. "Hopf algebras over commutative rings." In University Lecture Series, 53–62. Providence, Rhode Island: American Mathematical Society, 1999. http://dx.doi.org/10.1090/ulect/014/06.

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Chambert-Loir, Antoine. "Rings." In (Mostly) Commutative Algebra, 1–53. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-61595-6_1.

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Olberding, Bruce. "Finitely Stable Rings." In Commutative Algebra, 269–91. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0925-4_16.

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Fröberg, Ralf. "Stanley-Reisner Rings." In Commutative Algebra, 317–41. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89694-2_10.

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Peruginelli, Giulio, and Nicholas J. Werner. "Integral Closure of Rings of Integer-Valued Polynomials on Algebras." In Commutative Algebra, 293–305. New York, NY: Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0925-4_17.

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Rotman, Joseph. "Commutative rings I." In Advanced Modern Algebra, 81–171. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/gsm/114/02.

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Rotman, Joseph. "Commutative rings II." In Advanced Modern Algebra, 295–390. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/gsm/114/05.

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Rotman, Joseph. "Commutative rings III." In Advanced Modern Algebra, 873–983. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/gsm/114/10.

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Conference papers on the topic "Commutative Rings and Algebras"

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KLINGLER, LEE, and LAWRENCE S. LEVY. "REPRESENTATION TYPE OF COMMUTATIVE NOETHERIAN RINGS (INTRODUCTION)." In Proceedings of the International Conference on Algebras, Modules and Rings. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812774552_0010.

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aljohani, sarah. "A note on regularity conditions on Leavitt path algebras over commutative rings." In International Conference on Medical Imaging, Electronic Imaging, Information Technologies, and Sensors (MIEITS 2024), edited by Jyoti Jaiswal. SPIE, 2024. http://dx.doi.org/10.1117/12.3030774.

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Klisowski, Michal, and Vasyl Ustimenko. "On the implementation of public keys algorithms based on algebraic graphs over finite commutative rings." In 2010 International Multiconference on Computer Science and Information Technology (IMCSIT 2010). IEEE, 2010. http://dx.doi.org/10.1109/imcsit.2010.5679687.

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KRASNOV, YAKOV. "COMMUTATIVE ALGEBRAS IN CLIFFORD ANALYSIS." In Proceedings of the 3rd ISAAC Congress. World Scientific Publishing Company, 2003. http://dx.doi.org/10.1142/9789812794253_0041.

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Yinfeng Lu and Xiaohong Zhang. "Commutative pseudo-BCI algebras and commutative pseudo-BCI filters." In 2010 3rd International Symposium on Knowledge Acquisition and Modeling (KAM). IEEE, 2010. http://dx.doi.org/10.1109/kam.2010.5646322.

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Rachunek, Jiri, and Dana Salounova. "State operators on commutative basic algebras." In 2012 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). IEEE, 2012. http://dx.doi.org/10.1109/fuzz-ieee.2012.6251273.

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Al-Labadi, Manal, Shuker M. Alsalem, Enoch Suleiman, and Namuma Yerima. "On {1}-commutative permutation BP-algebras." In International Conference on Mathematical and Statistical Physics, Computational Science, Education and Communication (ICMSCE 2023), edited by Anton Purnama and Burhanuddin Arafah. SPIE, 2023. http://dx.doi.org/10.1117/12.3011417.

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Morier-Genoud, S., V. Ovsienko, Piotr Kielanowski, S. Twareque Ali, Anatol Odzijewicz, Martin Schlichenmaier, and Theodore Voronov. "Graded Commutative Algebras: Examples, Classification, Open Problems." In XXVIII WORKSHOP ON GEOMETRICAL METHODS IN PHYSICS. AIP, 2009. http://dx.doi.org/10.1063/1.3275586.

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Vasilevski, Nikolai, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "Commutative Algebras of Toeplitz Operators in Action." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3637748.

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Moldovyan, A. A., D. N. Moldovyan, May Thu Duong, Bac Thi Do, and Minh Hieu Nguyen. "Post-Quantum Signature Scheme on Commutative Algebras." In 2023 15th International Conference on Knowledge and Systems Engineering (KSE). IEEE, 2023. http://dx.doi.org/10.1109/kse59128.2023.10299530.

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Reports on the topic "Commutative Rings and Algebras"

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Yıldız, Eda, Ünsal Tekir, and Suat Koç. (2,J)-Ideals in Commutative Rings. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, September 2020. http://dx.doi.org/10.7546/crabs.2020.09.02.

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Kamen, Edward W. Control of Linear Systems Over Commutative Normed Algebras with Applications. Fort Belvoir, VA: Defense Technical Information Center, February 1987. http://dx.doi.org/10.21236/ada178765.

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