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Diskr. Mat., 1991, Volume 3, Issue 4, Pages 105–127 (Mi dm826)  

This article is cited in 30 scientific papers (total in 30 papers)

Linear recurrent sequences over commutative rings

A. A. Nechaev

Abstract: For a Noetherian commutative ring $\mathbf R$ with a unity there exist Galois correspondences between the structure of finitely generated submodules of the $R[x]$-module $\mathcal L_\mathbf R$ of all linear recurrent sequences (LRS) over $R$ and the structure of unitary ideals (the annihilators of these modules) in $R[x]$. We prove that these correspondences are one-to-one if and only if $R$ is a quasi-Frobenius ring. In this case we show that the well-known relations between sums and intersections of modules and their annihilators for LRS over fields are preserved. In the case when $R$ is also a principal ideal ring we construct a system of generators for the module of all LRS that are annihilated by a given unitary ideal, and derive a test for the cyclicity of this module over the ring $R[x]$.

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English version:
Discrete Mathematics and Applications, 1992, 2:6, 659–683

Bibliographic databases:
UDC: 621.391; 519.49
Received: 10.09.1990

Citation: A. A. Nechaev, “Linear recurrent sequences over commutative rings”, Diskr. Mat., 3:4 (1991), 105–127; Discrete Math. Appl., 2:6 (1992), 659–683

Citation in format AMSBIB
\by A.~A.~Nechaev
\paper Linear recurrent sequences over commutative rings
\jour Diskr. Mat.
\yr 1991
\vol 3
\issue 4
\pages 105--127
\jour Discrete Math. Appl.
\yr 1992
\vol 2
\issue 6
\pages 659--683

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    This publication is cited in the following articles:
    1. A. S. Kuz'min, A. A. Nechaev, “Linear recursive sequences over Galois rings”, Russian Math. Surveys, 48:1 (1993), 171–172  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. S. Kuz'min, “Lower estimates for the ranks of coordinate sequences of linear recurrent sequences over primary residue rings of integers”, Russian Math. Surveys, 48:3 (1993), 203–204  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. A. Nechaev, “Linear recurrent sequences over quasi-Frobenius modules”, Russian Math. Surveys, 48:3 (1993), 209–210  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. A. Nechaev, “Konechnye kvazifrobeniusovy moduli, prilozheniya k kodam i lineinym rekurrentam”, Fundament. i prikl. matem., 1:1 (1995), 229–254  mathnet  mathscinet  zmath  elib
    5. V. L. Kurakin, “Binomialnoe predstavlenie lineinykh rekurrentnykh posledovatelnostei”, Fundament. i prikl. matem., 1:2 (1995), 553–556  mathnet  mathscinet  zmath
    6. Kurakin V.L., Kuzmin A.S., Markov V.T., Mikhalev A.V., Nechaev A.A., “Linear codes and polylinear recurrences over finite rings and modules (a survey)”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings, Lecture Notes in Computer Science, 1719, 1999, 365–391  crossref  mathscinet  zmath  isi
    7. V. L. Kurakin, “Polynomial transformations of linear recurrent sequences over finite commutative rings”, Discrete Math. Appl., 10:4 (2000), 333–366  mathnet  crossref  mathscinet  zmath
    8. Lu P.Z., “A criterion for annihilating ideals of linear recurring sequences over Galois rings”, Applicable Algebra in Engineering Communication and Computing, 11:2 (2000), 141–156  crossref  mathscinet  isi
    9. Nechaev A.A., “Recurring sequences”, Formal Power Series and Algebraic Combinatorics, 2000, 54–66  isi
    10. A. A. Nechaev, D. A. Mikhailov, “A canonical system of generators of a unitary polynomial ideal over a commutative Artinian chain ring”, Discrete Math. Appl., 11:6 (2001), 545–586  mathnet  crossref  mathscinet  zmath
    11. Norton G.H., Salagean A., “Cyclic codes and minimal strong Grobner bases over a principal ideal ring”, Finite Fields and Their Applications, 9:2 (2003), 237–249  crossref  mathscinet  zmath  isi
    12. D. A. Mikhailov, A. A. Nechaev, “Solving systems of polynomial equations over Galois–Eisenstein rings with the use of the canonical generating systems of polynomial ideals”, Discrete Math. Appl., 14:1 (2004), 41–73  mathnet  crossref  crossref  mathscinet  zmath
    13. E. V. Gorbatov, “The standard basis of a polynomial ideal over a commutative Artinian chain ring”, Discrete Math. Appl., 14:1 (2004), 75–101  mathnet  crossref  crossref  mathscinet  zmath
    14. E. V. Gorbatov, “Standard bases concordant with the norm and computations in ideals and polylinear recurring sequences”, J. Math. Sci., 139:4 (2006), 6672–6707  mathnet  crossref  mathscinet  zmath
    15. Salagean A., “Repeated-root cyclic and negacyclic codes over a finite chain ring”, Discrete Applied Mathematics, 154:2 (2006), 413–419  crossref  mathscinet  zmath  isi
    16. E. V. Gorbatov, “Multiplicative orders on terms”, J. Math. Sci., 152:4 (2008), 517–521  mathnet  crossref  mathscinet  zmath
    17. Dinh H.Q., Lopez-Permouth S.R., Szabo S., “On the Structure of Cyclic and Negacyclic Codes over Finite Chain Rings”, Codes Over Rings, Series on Coding Theory and Cryptology, 6, 2009, 22–59  isi
    18. A. S. Kuzmin, G. B. Marshalko, A. A. Nechaev, “Vosstanovlenie lineinoi rekurrenty nad primarnym koltsom vychetov po ee uslozhneniyu”, Matem. vopr. kriptogr., 1:2 (2010), 31–56  mathnet  crossref
    19. Kuijper M., Schindelar K., “Minimal Grobner bases and the predictable leading monomial property”, Linear Algebra and Its Applications, 434:1 (2011), 104–116  crossref  zmath  isi
    20. A. S. Kuzmin, A. A. Nechaev, “Reconstruction of a linear recurrence of maximal period over a Galois ring from its highest coordinate sequence”, Discrete Math. Appl., 21:2 (2011), 145–178  mathnet  crossref  crossref  mathscinet  elib
    21. A. S. Kuzmin, G. B. Marshalko, “Vosstanovlenie lineinoi rekurrenty nad primarnym koltsom vychetov po ee uslozhneniyu. II”, Matem. vopr. kriptogr., 2:2 (2011), 81–93  mathnet  crossref
    22. M. A. Goltvanitsa, S. N. Zaitsev, A. A. Nechaev, “Skew linear recurring sequences of maximal period over Galois rings”, J. Math. Sci., 187:2 (2012), 115–128  mathnet  crossref
    23. Lopez-Permouth S.R., Ozadam H., Ozbudak F., Szabo S., “Polycyclic Codes Over Galois Rings with Applications to Repeated-Root Constacyclic Codes”, Finite Fields their Appl., 19:1 (2013), 16–38  crossref  isi
    24. A. V. Akishin, “On groups with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:4 (2015), 187–192  mathnet  crossref  crossref  mathscinet  isi  elib
    25. S. N. Zaitsev, “Description of maximal skew linear recurrences in terms of multipliers”, Matem. vopr. kriptogr., 5:2 (2014), 57–70  mathnet  crossref
    26. A. V. Akishin, “On groups of even orders with automorphisms generating recurrent sequences of the maximal period”, Discrete Math. Appl., 25:5 (2015), 253–259  mathnet  crossref  crossref  mathscinet  isi  elib
    27. V. N. Tsypyschev, “The second coordinate sequence of the MP-LRS over nontrivial Galois ring of an odd characteristic”, Discrete Math. Appl., 27:1 (2017), 35–54  mathnet  crossref  crossref  mathscinet  isi  elib
    28. V. N. Tsypyshev, “Hizhnie otsenki rangov koordinatnykh posledovatelnostei lineinykh rekurrent maksimalnogo perioda nad sobstvennym koltsom Galua”, Matem. vopr. kriptogr., 7:3 (2016), 137–143  mathnet  crossref  mathscinet  elib
    29. Kuijper M. Pinto R., “An iterative algorithm for parametrization of shortest length linear shift registers over finite chain rings”, Des. Codes Cryptogr., 83:2 (2017), 283–305  crossref  mathscinet  zmath  isi  scopus
    30. Tsypyschev V.N., “Lower Bounds on Linear Complexity of Digital Sequences Products of Lrs and Matrix Lrs Over Galois Ring”, Cybernetics Approaches in Intelligent Systems: Computational Methods in Systems and Software 2017, Vol. 1, Advances in Intelligent Systems and Computing, 661, ed. Silhavy R. Silhavy P. Prokopova Z., Springer International Publishing Ag, 2018, 50–61  crossref  isi
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