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Journal articles on the topic 'Rings (Algebra) Noether's theorem'

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Published: 4 June 2021

Last updated: 10 February 2022

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1

Osterburg, James, and Declan Quinn. "A Noether Skolem theorem for group-graded rings." Journal of Algebra 113, no. 2 (1988): 483–90. http://dx.doi.org/10.1016/0021-8693(88)90174-3.

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2

Lee†, Tsiu-Kwen, and Kun-Shan Liu. "The Skolem–Noether Theorem for Semiprime Rings Satisfying a Strict Identity." Communications in Algebra 35, no. 6 (2007): 1949–55. http://dx.doi.org/10.1080/00927870701247062.

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3

Osterburg, James, and Declan Quinn. "An addendum to a Noether Skolem theorem for group-graded rings." Journal of Algebra 120, no. 2 (1989): 414–15. http://dx.doi.org/10.1016/0021-8693(89)90205-6.

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4

FONTANA, M., P. JARA, and E. SANTOS. "PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS." Journal of Algebra and Its Applications 02, no. 01 (2003): 21–50. http://dx.doi.org/10.1142/s0219498803000349.

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Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter–Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J. M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer ]10] and B. G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. We also deal with the preservation of the P⋆MD property by ascent and descent in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

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5

Anderson, Ian M., and Juha Pohjanpelto. "Symmetries, conservation laws and variational principles for vector field theories†." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 2 (1996): 369–84. http://dx.doi.org/10.1017/s0305004100074922.

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The interplay between symmetries, conservation laws, and variational principles is a rich and varied one and extends well beyond the classical Noether's theorem. Recall that Noether's first theorem asserts that to every r dimensional Lie algebra of (generalized) symmetries of a variational problem there are r conserved quantities for the corresponding Euler-Lagrange equations. Noether's second theorem asserts that infinite dimensional symmetry algebras (depending upon arbitrary functions of all the independent variables) lead to differential identities for the Euler-Lagrange equations.

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6

Contiero, André, Lia Feital, and Renato Vidal Martins. "Max Noether's Theorem for integral curves." Journal of Algebra 494 (January 2018): 111–36. http://dx.doi.org/10.1016/j.jalgebra.2017.10.009.

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7

Brivio, Sonia, and Gian Pietro Pirola. "A Nonlinear Version of Noether's Type Theorem." Communications in Algebra 32, no. 7 (2004): 2723–32. http://dx.doi.org/10.1081/agb-120037412.

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8

Saworotnow, Parfeny P. "Gelfand theorem implies Stone representation theorem of Boolean rings." International Journal of Mathematics and Mathematical Sciences 18, no. 4 (1995): 701–4. http://dx.doi.org/10.1155/s0161171295000895.

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Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

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9

FAITH, CARL. "FACTOR RINGS OF PSEUDO-FROBENIUS RINGS." Journal of Algebra and Its Applications 05, no. 06 (2006): 847–54. http://dx.doi.org/10.1142/s0219498806001831.

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If R is right pseudo-Frobenius (= PF), and A is an ideal, when is R/A right PF? Our main result, Theorem 3.7, states that this happens iff the ideal A′ of the basic ring B of R corresponding to A has left annihilator F in B generated by a single element on both sides. Moreover, in this case B/A′ ≈ F in mod-B, (see Theorem 3.5), a property that does not extend to R, that is, in general R/A is not isomorphic to the left annihilator of A. (See Example 4.3(2) and Theorem 4.5.) Theorem 4.6 characterizes Frobenius rings among quasi-Frobenius (QF) rings. As an application of the main theorem, in Theorem 3.9 we prove that if A is generated as a right or left ideal by an idempotent e, then e is central (and R/A is then trivially right PF along with R). This generalizes the result of F. W. Anderson for quasi-Frobenius rings. (See Theorem 2.2 for a new proof.). In Proposition 1.6, we prove that a generalization of this result holds for finite products R of full matrix rings over local rings; namely, an ideal A is finitely generated as a right or left ideal iff A is generated by a central idempotent. We also note a theorem going back to Nakayama, Goursaud, and the author that every factor ring of R is right PF iff R is a uniserial ring. (See Theorem 5.1.).

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10

Almeida, Marcela, Manuela Blaum, Lisi D'Alfonso, and Pablo Solernó. "Computing bases of complete intersection rings in Noether position." Journal of Pure and Applied Algebra 162, no. 2-3 (2001): 127–70. http://dx.doi.org/10.1016/s0022-4049(00)00135-3.

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11

Xu, Jinzhong. "A theorem on artinian rings." Communications in Algebra 22, no. 3 (1994): 905–14. http://dx.doi.org/10.1080/00927879408824884.

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12

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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13

Quang Dinh, Hai, and Dinh Van Huynh. "A decomposition theorem for ℘∗-semisimple rings." Journal of Pure and Applied Algebra 186, no. 2 (2004): 139–49. http://dx.doi.org/10.1016/s0022-4049(03)00127-0.

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14

Larson, Suzanne. "THE INTERMEDIATE VALUE THEOREM INf-RINGS." Communications in Algebra 30, no. 5 (2002): 2469–504. http://dx.doi.org/10.1081/agb-120003479.

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15

Frederico, Gastão, and Delfim Torres. "Noether's symmetry Theorem for variational and optimal control problems with time delay." Numerical Algebra, Control and Optimization 2, no. 3 (2012): 619–30. http://dx.doi.org/10.3934/naco.2012.2.619.

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16

UPADHYAY, SUDHAKER, and BHABANI PRASAD MANDAL. "NONCOMMUTATIVE GAUGE THEORIES: MODEL FOR HODGE THEORY." International Journal of Modern Physics A 28, no. 25 (2013): 1350122. http://dx.doi.org/10.1142/s0217751x13501224.

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The nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST, dual-BRST and anti-dual-BRST symmetry transformations are constructed in the context of noncommutative (NC) 1-form as well as 2-form gauge theories. The corresponding Noether's charges for these symmetries on the Moyal plane are shown to satisfy the same algebra, as by the de Rham cohomological operators of differential geometry. The Hodge decomposition theorem on compact manifold is also studied. We show that noncommutative gauge theories are field theoretic models for Hodge theory.

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17

Tang, Gaohua, and Yiqiang Zhou. "An embedding theorem on triangular matrix rings." Linear and Multilinear Algebra 65, no. 5 (2016): 882–90. http://dx.doi.org/10.1080/03081087.2016.1211083.

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18

Tan, Yi-Jia. "Fillmore's theorem for matrices over factorial rings." Linear and Multilinear Algebra 68, no. 3 (2018): 563–67. http://dx.doi.org/10.1080/03081087.2018.1509045.

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19

Barioli, Francesco, Alberto Facchini, Francisco Raggi, and José Ríos. "KRULL-SCHMIDT THEOREM AND HOMOGENEOUS SEMILOCAL RINGS." Communications in Algebra 29, no. 4 (2001): 1649–58. http://dx.doi.org/10.1081/agb-100002124.

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20

Der Kallen, Wilberd Van. "Vaserstein's pre-stabilization theorem over commutative rings." Communications in Algebra 15, no. 3 (1987): 657–63. http://dx.doi.org/10.1080/00927878708823435.

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21

Oda, Fumihito, and Tomoyuki Yoshida. "Crossed Burnside Rings I. The Fundamental Theorem." Journal of Algebra 236, no. 1 (2001): 29–79. http://dx.doi.org/10.1006/jabr.2000.8341.

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22

Radjabalipour, M., P. Rosenthal, and B. R. Yahaghi. "Burnside's theorem for matrix rings over division rings." Linear Algebra and its Applications 383 (May 2004): 29–44. http://dx.doi.org/10.1016/j.laa.2003.08.011.

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23

Lashkhi, A., and T. Kvirikashvili. "On the Fundamental Theorem of Geometric Algebra Over SF-Rings." Communications in Algebra 36, no. 9 (2008): 3564–73. http://dx.doi.org/10.1080/00927870701776557.

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24

Gamanda, Maroua, Henri Lombardi, Stefan Neuwirth, and Ihsen Yengui. "The syzygy theorem for Bézout rings." Mathematics of Computation 89, no. 322 (2019): 941–64. http://dx.doi.org/10.1090/mcom/3466.

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25

Dugas, Manfred, and Rüdiger Göbel. "An extension of Zassenhaus' theorem on endomorphism rings." Fundamenta Mathematicae 194, no. 3 (2007): 239–51. http://dx.doi.org/10.4064/fm194-3-2.

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26

Shimomoto, Kazuma, and Wenliang Zhang. "On the localization theorem for F-pure rings." Journal of Pure and Applied Algebra 213, no. 6 (2009): 1133–39. http://dx.doi.org/10.1016/j.jpaa.2008.11.047.

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27

Faith, Carl. "Dedekind Finite Rings and a Theorem of Kaplansky." Communications in Algebra 31, no. 9 (2003): 4175–78. http://dx.doi.org/10.1081/agb-120022785.

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28

Banaschewski, B. "The prime ideal theorem and representation ofF-rings." Algebra Universalis 25, no. 1 (1988): 384–87. http://dx.doi.org/10.1007/bf01229983.

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29

Nicholson, W. K., and E. Sánchez Campos. "Rings with the dual of the isomorphism theorem." Journal of Algebra 271, no. 1 (2004): 391–406. http://dx.doi.org/10.1016/j.jalgebra.2002.10.001.

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30

Kleinert, Ernst. "A theorem on units of integral group rings." Journal of Pure and Applied Algebra 49, no. 1-2 (1987): 161–71. http://dx.doi.org/10.1016/0022-4049(87)90126-5.

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31

Borceux, F., and R. Cruciani. "A Generic Representation Theorem for Non-commutative Rings." Journal of Algebra 167, no. 2 (1994): 291–308. http://dx.doi.org/10.1006/jabr.1994.1186.

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32

Liu, Gongxiang, and Fang Li. "On Strongly Groupoid Graded Rings and the Corresponding Clifford Theorem." Algebra Colloquium 13, no. 02 (2006): 181–96. http://dx.doi.org/10.1142/s1005386706000198.

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In this paper, we introduce the definition of groupoid graded rings. Group graded rings, (skew) groupoid rings, artinian semisimple rings, matrix rings and others can be regarded as special kinds of groupoid graded rings. Our main task is to classify strongly groupoid graded rings by cohomology of groupoids. Some classical results about group graded rings are generalized to groupoid graded rings. In particular, the Clifford Theorem for a strongly groupoid graded ring is given.

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33

Gouvei, Paulo D. F., and Delfim F. M. Torres. "Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control." Computational Methods in Applied Mathematics 5, no. 4 (2005): 387–409. http://dx.doi.org/10.2478/cmam-2005-0018.

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AbstractWe present analytical computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples are given.

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34

Kelly, Shane, and Matthew Morrow. "K-theory of valuation rings." Compositio Mathematica 157, no. 6 (2021): 1121–42. http://dx.doi.org/10.1112/s0010437x21007119.

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We prove several results showing that the algebraic $K$-theory of valuation rings behaves as though such rings were regular Noetherian, in particular an analogue of the Geisser–Levine theorem. We also give some new proofs of known results concerning cdh descent of algebraic $K$-theory.

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35

Fernandez Lopez,, A., E. Garcia Rus, and E. Sanchez Campos. "Structure theorem for prime rings satisfying a generalized identity." Communications in Algebra 22, no. 5 (1994): 1729–40. http://dx.doi.org/10.1080/00927879408824932.

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36

Chuang #, Chen-Lian, and Cheng-Kai Liu. "Extended Jacobson Density Theorem for Rings With Skew Derivations." Communications in Algebra 35, no. 4 (2007): 1391–413. http://dx.doi.org/10.1080/00927870601142207.

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37

Mulcahy, Colm. "A representation theorem for higher level reduced Witt rings." Journal of Algebra 119, no. 1 (1988): 105–22. http://dx.doi.org/10.1016/0021-8693(88)90078-6.

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38

Generalov, A. I., and I. M. Zilberbord. "Generalized “stacked bases” theorem for modules over semiperfect rings." Communications in Algebra 49, no. 6 (2021): 2597–605. http://dx.doi.org/10.1080/00927872.2021.1879105.

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39

de Seguins Pazzis, Clément. "The Flanders theorem over division rings." Linear Algebra and its Applications 493 (March 2016): 313–22. http://dx.doi.org/10.1016/j.laa.2015.11.022.

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40

Wang, Zhou, and Jianlong Chen. "A Note on Clean Rings." Algebra Colloquium 14, no. 03 (2007): 537–40. http://dx.doi.org/10.1142/s1005386707000491.

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Let R be a ring and g(x) a polynomial in C[x], where C=C(R) denotes the center of R. Camillo and Simón called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x). In this paper, we prove that for a, b ∈ C, the ring R is clean and b-a is invertible in R if and only if R is g1(x)-clean, where g1(x)=(x-a)(x-b). This implies that in some sense the notion of g(x)-clean rings in the Nicholson–Zhou Theorem and in the Camillo–Simón Theorem is indeed equivalent to the notion of clean rings.

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41

George, Seleena M., Roy L. McCasland, and Patrick F. Smith. "A principal ideal theorem analogue for modules over commutative rings." Communications in Algebra 22, no. 6 (1994): 2083–99. http://dx.doi.org/10.1080/00927879408824957.

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42

Hindman, Peter, Lee Klingler, and Charles J. Odenthal. "On the krull-schmidt-azumaya theorem for integral group rings." Communications in Algebra 26, no. 11 (1998): 3743–58. http://dx.doi.org/10.1080/00927879808826371.

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43

Osmanagic, E. "On an approximation theorem for krull rings with zero divisors." Communications in Algebra 27, no. 8 (1999): 3647–57. http://dx.doi.org/10.1080/00927879908826653.

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44

Chen, Cao-yu, and Warren D. Nichols. "A duality theorem for hope module algebras over dedekind rings." Communications in Algebra 18, no. 10 (1990): 3209–21. http://dx.doi.org/10.1080/00927879008824070.

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45

Barhoumi, Sami, and Henri Lombardi. "An algorithm for the Traverso–Swan theorem on seminormal rings." Journal of Algebra 320, no. 4 (2008): 1531–42. http://dx.doi.org/10.1016/j.jalgebra.2008.05.013.

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46

Garcı́a, J. L., and L. Marı́n. "An Extension of a Theorem on Endomorphism Rings and Equivalences." Journal of Algebra 181, no. 3 (1996): 962–66. http://dx.doi.org/10.1006/jabr.1996.0156.

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47

Lee, Tsiu-Kwen. "Ad-nilpotent Elements of Semiprime Rings with Involution." Canadian Mathematical Bulletin 61, no. 2 (2018): 318–27. http://dx.doi.org/10.4153/cmb-2017-005-3.

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AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.

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48

Fernando, José F. "ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS." Journal of the Institute of Mathematics of Jussieu 14, no. 4 (2014): 857–94. http://dx.doi.org/10.1017/s1474748014000206.

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Let$R\subset F$be an extension of real closed fields, and let${\mathcal{S}}(M,R)$be the ring of (continuous) semialgebraic functions on a semialgebraic set$M\subset R^{n}$. We prove that every$R$-homomorphism${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$is essentially the evaluation homomorphism at a certain point$p\in F^{n}$adjacent to the extended semialgebraic set$M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when$M$is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

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49

Wang, Fanggui, and Lei Qiao. "Two applications of Nagata rings and modules." Journal of Algebra and Its Applications 19, no. 06 (2019): 2050115. http://dx.doi.org/10.1142/s0219498820501157.

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Let [Formula: see text] be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China 10 (2015) 91–110]. First they are used to obtain Chase’s Theorem for [Formula: see text]-coherent rings. In particular, we obtain the [Formula: see text]-version of Chase’s Theorem, where [Formula: see text] is the classical star operation in ideal theory. In the second half, we apply they to characterize [Formula: see text]-flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)].

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50

Ánh, P. N., and L. Márki. "Orders in primitive rings with non-zero socle and posner's theorem." Communications in Algebra 24, no. 1 (1996): 289–94. http://dx.doi.org/10.1080/00927879608825567.

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