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Journal articles on the topic 'Galois ring'

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Published: 4 June 2021

Last updated: 11 February 2022

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1

Kozlitin, Oleg A. "Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring." Discrete Mathematics and Applications 28, no. 6 (December 19, 2018): 345–58. http://dx.doi.org/10.1515/dma-2018-0031.

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Abstract The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.

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2

Szeto, George. "Separable subalgebras of a class of Azumaya algebras." International Journal of Mathematics and Mathematical Sciences 21, no. 2 (1998): 235–38. http://dx.doi.org/10.1155/s0161171298000337.

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LetSbe a ring with 1,Cthe center ofS,Ga finite automorphism group ofSof orderninvertible inS, andSGthe subnng of elements ofSfixed under each element inG. It is shown that the skew group ringS*Gis aG′-Galois extension of(S*G)G′that is a projective separableCG-algebra whereG′is the inner automorphism group ofS*Ginduced byGif and only ifSis aG-Galois extension ofSGthat is a projective separableCG-algebra. Moreover, properties of the separable subalgebras of aG-GaloisH-separable extensionSofSGare given whenSGis a projective separableCG-algebra.

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3

Szeto, George, and Lianyong Xue. "On central commutator Galois extensions of rings." International Journal of Mathematics and Mathematical Sciences 24, no. 5 (2000): 289–94. http://dx.doi.org/10.1155/s0161171200004099.

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LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

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4

Zhang, Aixian, and Keqin Feng. "Normal Bases on Galois Ring Extensions." Symmetry 10, no. 12 (December 3, 2018): 702. http://dx.doi.org/10.3390/sym10120702.

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Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

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5

Szeto, George, and Linjun Ma. "On Galois projective group rings." International Journal of Mathematics and Mathematical Sciences 14, no. 1 (1991): 149–53. http://dx.doi.org/10.1155/s0161171291000145.

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LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {UαinA/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

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6

Szeto, George. "On Azumaya Galois extensions and skew group rings." International Journal of Mathematics and Mathematical Sciences 22, no. 1 (1999): 91–95. http://dx.doi.org/10.1155/s0161171299220911.

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Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

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7

PAQUES, ANTONIO, VIRGÍNIA RODRIGUES, and ALVERI SANT'ANA. "GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS." Journal of Algebra and Its Applications 10, no. 05 (October 2011): 835–47. http://dx.doi.org/10.1142/s0219498811004999.

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Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

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8

Mináč, Ján, and Tara L. Smith. "Decomposition of Witt Rings and Galois Groups." Canadian Journal of Mathematics 47, no. 6 (December 1, 1995): 1274–89. http://dx.doi.org/10.4153/cjm-1995-065-0.

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AbstractTo each field F of characteristic not 2, one can associate a certain Galois group 𝒢F, the so-called W-group of F, which carries essentially the same information as the Witt ring W(F) of F. In this paper we show that direct products of Witt rings correspond to free products of these Galois groups (in the appropriate category), group ring construction of Witt rings corresponds to semidirect products of W-groups, and the basic part of W(F) is related to the center of 𝒢F. In an appendix we provide a complete list of Witt rings and corresponding w-groups for fields F with |Ḟ/Ḟ2| ≤ 16.

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9

Jiang, Xiaolong, and George Szeto. "The Galois endomorphism ring of a Galois Azumaya extension." International Journal of Algebra 7 (2013): 909–14. http://dx.doi.org/10.12988/ija.2013.311111.

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10

Cao, Yonglin. "Generalized affine transformation monoids on Galois rings." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–6. http://dx.doi.org/10.1155/ijmms/2006/90738.

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LetAbe a ring with identity. The generalized affine transformation monoidGaff(A)is defined as the set of all transformations onAof the formx↦xu+a(for allx∈A), whereu,a∈A. We study the algebraic structure of the monoidGaff(A)on a finite Galois ringA. The following results are obtained: an explicit description of Green's relations onGaff(A); and an explicit description of the Schützenberger group of every-class, which is shown to be isomorphic to the affine transformation group for a smaller Galois ring.

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11

Szeto, George, and Lianyong Xue. "Skew group rings which are Galois." International Journal of Mathematics and Mathematical Sciences 23, no. 4 (2000): 279–83. http://dx.doi.org/10.1155/s0161171200000624.

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LetS*Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS*Gis anG′-Galois extension of(S*G)G′, whereG′is the inner automorphism group ofS*Ginduced by the elements inG, thenSis aG-Galois extension ofSG. A necessary and sufficient condition is also given for the commutator subring of(S*G)G′inS*Gto be a Galois extension, where(S*G)G′is the subring of the elements fixed under each element inG′.

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12

Ku-Cauich, Juan, Guillermo Morales-Luna, and Horacio Tapia-Recillas. "An Authentication Code over Galois Rings with Optimal Impersonation and Substitution Probabilities." Mathematical and Computational Applications 23, no. 3 (September 6, 2018): 46. http://dx.doi.org/10.3390/mca23030046.

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Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned to the existence of a special class of bent maps on Galois rings.

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13

Evdokimov, Sergei, and Ilya Ponomarenko. "Coset closure of a circulant S-ring and schurity problem." Journal of Algebra and Its Applications 15, no. 04 (February 19, 2016): 1650068. http://dx.doi.org/10.1142/s0219498816500687.

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Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

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14

Nakajima, Atsushi. "Weak hopf galois extensions andp-galois extensions of a ring." Communications in Algebra 23, no. 8 (January 1995): 2851–62. http://dx.doi.org/10.1080/00927879508825372.

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15

Bouc, Serge. "The slice Burnside ring and the section Burnside ring of a finite group." Compositio Mathematica 148, no. 3 (March 22, 2012): 868–906. http://dx.doi.org/10.1112/s0010437x11007500.

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AbstractThis paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets and of Galois morphisms of G-sets, respectively. The well-known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural Green biset functor structure. The functorial structure of unit groups of these rings is also discussed.

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16

Evdokimov, Sergei, and Ilya Ponomarenko. "Schur rings over a Galois ring of odd characteristic." Journal of Combinatorial Theory, Series A 117, no. 7 (October 2010): 827–41. http://dx.doi.org/10.1016/j.jcta.2009.12.008.

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17

Ho, Wei, and Matthew Satriano. "Galois Closures of Non-commutative Rings and an Application to Hermitian Representations." International Mathematics Research Notices 2020, no. 21 (October 3, 2018): 7944–74. http://dx.doi.org/10.1093/imrn/rny231.

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Abstract Galois closures of commutative rank $n$ ring extensions were introduced by Bhargava and the 2nd author. In this paper, we generalize the construction to the case of non-commutative rings. We show that noncommutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.

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18

SATO, JUNRO, SUSUMU ODA, and KEN-ICHI YOSHIDA. "ON INTEGRAL DOMAINS WITH CYCLIC GROUP ACTIONS." Journal of Algebra and Its Applications 10, no. 03 (June 2011): 491–508. http://dx.doi.org/10.1142/s0219498811004719.

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Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization [Formula: see text] of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

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19

Singh, Abhay Kumar, Narendra Kumar, and Kar Ping Shum. "Cyclic self-orthogonal codes over finite chain ring." Asian-European Journal of Mathematics 11, no. 06 (December 2018): 1850078. http://dx.doi.org/10.1142/s179355711850078x.

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In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq. 6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over [Formula: see text].

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20

HASHEMI, AMIR, and PARISA ALVANDI. "APPLYING BUCHBERGER'S CRITERIA FOR COMPUTING GRÖBNER BASES OVER FINITE-CHAIN RINGS." Journal of Algebra and Its Applications 12, no. 07 (May 16, 2013): 1350034. http://dx.doi.org/10.1142/s0219498813500345.

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Norton and Sălăgean [Strong Gröbner bases and cyclic codes over a finite-chain ring, in Proc. Workshop on Coding and Cryptography, Paris, Electronic Notes in Discrete Mathematics, Vol. 6 (Elsevier Science, 2001), pp. 391–401] have presented an algorithm for computing Gröbner bases over finite-chain rings. Byrne and Fitzpatrick [Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput.31 (2001) 565–584] have simultaneously proposed a similar algorithm for computing Gröbner bases over Galois rings (a special kind of finite-chain rings). However, they have not incorporated Buchberger's criteria into their algorithms to avoid unnecessary reductions. In this paper, we propose the adapted version of these criteria for polynomials over finite-chain rings and we show how to apply them on Norton–Sălăgean algorithm. The described algorithm has been implemented in Maple and experimented with a number of examples for the Galois rings.

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21

Szeto, George, and Lianyong Xue. "On characterizations of a center Galois extension." International Journal of Mathematics and Mathematical Sciences 23, no. 11 (2000): 753–58. http://dx.doi.org/10.1155/s0161171200003562.

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LetBbe a ring with1, Cthe center ofB, Ga finite automorphism group ofB, andBGthe set of elements inBfixed under each element inG. Then, it is shown thatBis a center Galois extension ofBG(that is,Cis a Galois algebra overCGwith Galois groupG|C≅G) if and only if the ideal ofBgenerated by{c−g(c)|c∈C}isBfor eachg≠1inG. This generalizes the well known characterization of a commutative Galois extensionCthatCis a Galois extension ofCGwith Galois groupGif and only if the ideal generated by{c−g(c)|c∈C}isCfor eachg≠1inG. Some more characterizations of a center Galois extensionBare also given.

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22

Szeto, George, and Lianyong Xue. "On weak center Galois extensions of rings." International Journal of Mathematics and Mathematical Sciences 25, no. 7 (2001): 489–95. http://dx.doi.org/10.1155/s016117120100504x.

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LetBbe a ring with 1,Cthe center ofB,Ga finite automorphism group ofB, andBGthe set of elements inBfixed under each element inG. Then, the notion of a center Galois extension ofBGwith Galois groupG(i.e.,Cis a Galois algebra overCGwith Galois groupG|C≅G) is generalized to a weak center Galois extension with groupG, whereBis called a weak center Galois extension with groupGifBIi=Beifor some idempotent inCandIi={c−gi(c)|c∈C}for eachgi≠1inG. It is shown thatBis a weak center Galois extension with groupGif and only if for eachgi≠1inGthere exists an idempotenteiinCand{bkei∈Bei;ckei∈Cei,k=1,2,...,m}such that∑k=1mbkeigi(ckei)=δ1,gieiandgirestricted toC(1−ei)is an identity, and a structure of a weak center Galois extension with groupGis also given.

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23

GONZÁLEZ, S., C. MARTÍNEZ, I. F. RÚA, V. T. MARKOV, and A. A. NECHAEV. "COORDINATE SETS OF GENERALIZED GALOIS RINGS." Journal of Algebra and Its Applications 03, no. 01 (March 2004): 31–48. http://dx.doi.org/10.1142/s0219498804000678.

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A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.

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24

Fernández-Alonso, Rogelio, and Janeth Magaña. "Galois connections between lattices of preradicals induced by ring epimorphisms." Journal of Algebra and Its Applications 19, no. 03 (February 27, 2019): 2050045. http://dx.doi.org/10.1142/s0219498820500450.

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We continue the study of Galois connections between the lattices of preradicals of two rings [Formula: see text] and [Formula: see text] induced by an adjoint pair of functors between the categories [Formula: see text]-Mod and [Formula: see text]-Mod. In this paper, we focus on the functor triple induced by any ring homomorphism [Formula: see text], and particularly when it is a ring epimorphism. We give additional results when the epimorphism is flat and when it is projective.

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25

Nuss, Philippe. "Galois-Azumaya extensions and the Brauer-Galois group of a commutative ring." Bulletin of the Belgian Mathematical Society - Simon Stevin 13, no. 2 (June 2006): 247–70. http://dx.doi.org/10.36045/bbms/1148059461.

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26

Chen, Weining, Gaohua Tang, and Huadong Su. "Units on the Gauss extension of a Galois ring." Journal of Algebra and Its Applications 15, no. 02 (October 6, 2015): 1650028. http://dx.doi.org/10.1142/s0219498816500286.

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Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group.

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27

Gepner, David, and Tyler Lawson. "Brauer groups and Galois cohomology of commutative ring spectra." Compositio Mathematica 157, no. 6 (June 2021): 1211–64. http://dx.doi.org/10.1112/s0010437x21007065.

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In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$, we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $\pi _0(E)$, recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$-algebra with the same coefficient ring as $\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$, previously studied by Mathew and Stojanoska.

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28

Halupczok, Immanuel. "Motives for perfect PAC fields with pro-cyclic Galois group." Journal of Symbolic Logic 73, no. 3 (September 2008): 1036–50. http://dx.doi.org/10.2178/jsl/1230396764.

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AbstractDenef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

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29

Lundström, Patrik. "Normal bases for infinite Galois ring extensions." Colloquium Mathematicum 79, no. 2 (1999): 235–40. http://dx.doi.org/10.4064/cm-79-2-235-240.

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30

Vasin, A. R. "Piecewise Polynomial Sequences over the Galois Ring." Problems of Information Transmission 56, no. 1 (January 2020): 91–102. http://dx.doi.org/10.1134/s0032946020010081.

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31

Labuz, M., and T. Lulek. "Magnetic Pentagonal Ring - Galois Extensions and Crystallography." Acta Physica Polonica A 132, no. 1 (July 2017): 97–99. http://dx.doi.org/10.12693/aphyspola.132.97.

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32

Zhang, XiaoLei, and Lei Hu. "Periods of polynomials over a Galois ring." Science China Mathematics 56, no. 9 (March 9, 2013): 1761–72. http://dx.doi.org/10.1007/s11425-013-4592-2.

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33

Irwansyah, Intan Muchtadi-Alamsyah, Aleams Barra, and Ahmad Muchlis. "Self-Dual Normal Basis of a Galois Ring." Journal of Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/258187.

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LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.

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34

LAU, EIKE. "DISPLAYED EQUATIONS FOR GALOIS REPRESENTATIONS." Nagoya Mathematical Journal 235 (February 13, 2018): 86–114. http://dx.doi.org/10.1017/nmj.2018.3.

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The Galois representation associated to a $p$ -divisible group over a normal complete noetherian local ring with perfect residue field is described in terms of its Dieudonné display. As a consequence, the Kisin module associated to a commutative finite flat $p$ -group scheme via Dieudonné displays is related to its Galois representation in the expected way.

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35

Norton, Graham H., and Ana Sǎlǎgean. "Strong Gröbner bases for polynomials over a principal ideal ring." Bulletin of the Australian Mathematical Society 64, no. 3 (December 2001): 505–28. http://dx.doi.org/10.1017/s0004972700019973.

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Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.

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36

Dobbs, David E. "A minimal ring extension of a large finite local prime ring is probably ramified." Journal of Algebra and Its Applications 19, no. 01 (January 29, 2019): 2050015. http://dx.doi.org/10.1142/s0219498820500152.

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Given any minimal ring extension [Formula: see text] of finite fields, several families of examples are constructed of a finite local (commutative unital) ring [Formula: see text] which is not a field, with a (necessarily finite) inert (minimal ring) extension [Formula: see text] (so that [Formula: see text] is a separable [Formula: see text]-algebra), such that [Formula: see text] is not a Galois extension and the residue field of [Formula: see text] (respectively, [Formula: see text]) is [Formula: see text] (respectively, [Formula: see text]). These results refute an assertion of G. Ganske and McDonald stating that if [Formula: see text] are finite local rings such that [Formula: see text] is a separable [Formula: see text]-algebra, then [Formula: see text] is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let [Formula: see text] be a finite special principal ideal ring (SPIR), but not a field, such that [Formula: see text] has index of nilpotency [Formula: see text] ([Formula: see text]). Impose the uniform distribution on the (finite) set of ([Formula: see text]-algebra) isomorphism classes of the minimal ring extensions of [Formula: see text]. If [Formula: see text] (for instance, if [Formula: see text]), the probability that a random isomorphism class consists of ramified extensions of [Formula: see text] is at least [Formula: see text]; if [Formula: see text] (for instance, if [Formula: see text] for some odd prime [Formula: see text]), the corresponding probability is at least [Formula: see text]. Additional applications, examples and historical remarks are given.

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37

Shah, Tariq, Attiq Qamar, and Iqtadar Hussain. "Substitution Box on Maximal Cyclic Subgroup of Units of a Galois Ring." Zeitschrift für Naturforschung A 68, no. 8-9 (September 1, 2013): 567–72. http://dx.doi.org/10.5560/zna.2013-0021.

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In this paper, we construct a new substitution box (S-box) structure based on the elements of the maximal cyclic subgroup of the multiplicative group of units in a finite Galois ring instead of Galois field. We analyze the potency of the proposed S-box by using the majority logic criterion. Moreover, we illustrate the utility of the projected S-box in watermarking.

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38

Nelson, Audrey, and George Szeto. "When is the ring of 2x2 matrices over a ring Galois?" International Journal of Algebra 7 (2013): 439–44. http://dx.doi.org/10.12988/ija.2013.3445.

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39

Ramakrishna, Ravi. "Constructing Galois Representations with Very Large Image." Canadian Journal of Mathematics 60, no. 1 (February 1, 2008): 208–21. http://dx.doi.org/10.4153/cjm-2008-009-7.

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AbstractStarting with a 2-dimensional mod p Galois representation, we construct a deformation to a power series ring in infinitely many variables over the p-adics. The image of this representation is full in the sense that it contains SL2 of this power series ring. Furthermore, all Zp specializations of this deformation are potentially semistable at p.

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40

Ankur. "Self-dual codes over the ring GR(pm,g) and Jacobi forms." Asian-European Journal of Mathematics 10, no. 03 (September 2017): 1750055. http://dx.doi.org/10.1142/s1793557117500553.

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We take a Galois ring [Formula: see text] and discuss about the self-dual codes and its properties over the ring. We will also describe the relationship between Clifford-Weil group and Jacobi forms by constructing the invariant polynomial ring with the complete weight enumerator.

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41

Burns, David, and Henri Johnston. "A non-abelian Stickelberger theorem." Compositio Mathematica 147, no. 1 (July 1, 2010): 35–55. http://dx.doi.org/10.1112/s0010437x10004859.

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AbstractLet L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

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42

Ara, Pere. "On the symmetric algebra of quotients of a C*-algebra." Glasgow Mathematical Journal 32, no. 3 (September 1990): 377–79. http://dx.doi.org/10.1017/s0017089500009460.

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Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

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Byott, N. P. "Some self-dual local rings of integers not free over their associated orders." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 1 (July 1991): 5–10. http://dx.doi.org/10.1017/s0305004100070067.

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Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

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Zaets, M. V. "Digit-polynomial construction of substitutions over galois ring." Prikladnaya diskretnaya matematika. Prilozhenie, no. 10 (September 1, 2017): 17–19. http://dx.doi.org/10.17223/2226308x/10/5.

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45

López-Díaz, M. C., and I. F. Rúa. "Witt’s theorems for Galois Ring valued quadratic forms." Journal of Pure and Applied Algebra 212, no. 11 (November 2008): 2493–502. http://dx.doi.org/10.1016/j.jpaa.2008.03.028.

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Arason, J{ón Kr, Richard Elman, and Bill Jacob. "The graded Witt ring and Galois cohomology. II." Transactions of the American Mathematical Society 314, no. 2 (February 1, 1989): 745. http://dx.doi.org/10.1090/s0002-9947-1989-0964897-9.

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47

Jiang, Xiaolong, and George Szeto. "The endomorphism ring of a Galois Azumaya extension." International Journal of Algebra 7 (2013): 527–32. http://dx.doi.org/10.12988/ija.2013.29110.

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48

López-Díaz, M. C., and I. F. Rúa. "Witt index for Galois Ring valued quadratic forms." Finite Fields and Their Applications 16, no. 3 (May 2010): 175–86. http://dx.doi.org/10.1016/j.ffa.2010.02.003.

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Szeto, George, and Lianyong Xue. "The general Ikehata theorem forH-separable crossed products." International Journal of Mathematics and Mathematical Sciences 23, no. 10 (2000): 657–62. http://dx.doi.org/10.1155/s0161171200003124.

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LetBbe a ring with1, Cthe center ofB, Gan automorphism group ofBof ordernfor some integern, CGthe set of elements inCfixed underG, Δ=Δ(B,G,f)a crossed product overBwherefis a factor set fromG×GtoU(CG). It is shown thatΔis anH-separable extension ofBandVΔ(B)is a commutative subring ofΔif and only ifCis a Galois algebra overCGwith Galois groupG|C≅G.

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50

Acosta-De-Orozco, Maria T., and Javier Gomez-Calderon. "Matrix powers over finite fields." International Journal of Mathematics and Mathematical Sciences 15, no. 4 (1992): 767–71. http://dx.doi.org/10.1155/s0161171292000991.

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LetGF(q)denote the finite field of orderq=pewithpodd. LetMdenote the ring of2×2matrices with entries inGF(q). Letndenote a divisor ofq−1and assume2≤nand4does not dividen. In this paper, we consider the problem of determining the number ofn-th roots inMof a matrixB∈M. Also, as a related problem, we consider the problem of lifting the solutions ofX2=Bover Galois rings.

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