Characterization of irreducible polynomials over a special principal ideal ring

Brahim Boudine

Mathematica Bohemica (2023)

  • Volume: 148, Issue: 4, page 501-506
  • ISSN: 0862-7959

Abstract

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A commutative ring with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length . Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length .

How to cite

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Boudine, Brahim. "Characterization of irreducible polynomials over a special principal ideal ring." Mathematica Bohemica 148.4 (2023): 501-506. <http://eudml.org/doc/299473>.

@article{Boudine2023,
abstract = {A commutative ring $R$ with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length $e$ is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length $2$. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length $e$.},
author = {Boudine, Brahim},
journal = {Mathematica Bohemica},
keywords = {polynomial; irreducibility; commutative principal ideal ring},
language = {eng},
number = {4},
pages = {501-506},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Characterization of irreducible polynomials over a special principal ideal ring},
url = {http://eudml.org/doc/299473},
volume = {148},
year = {2023},
}

TY - JOUR
AU - Boudine, Brahim
TI - Characterization of irreducible polynomials over a special principal ideal ring
JO - Mathematica Bohemica
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 148
IS - 4
SP - 501
EP - 506
AB - A commutative ring $R$ with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length $e$ is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length $2$. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length $e$.
LA - eng
KW - polynomial; irreducibility; commutative principal ideal ring
UR - http://eudml.org/doc/299473
ER -

References

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  1. Azumaya, G., 10.1017/S0027763000010114, Nagoya Math. J. 2 (1951), 119-150. (1951) Zbl0045.01103MR0040287DOI10.1017/S0027763000010114
  2. Berlekamp, E. R., 10.1142/9407, McGraw-Hill, New York (1968). (1968) Zbl0988.94521MR0238597DOI10.1142/9407
  3. Brown, W. C., Matrices Over Commutative Rings, Pure and Applied Mathematics 169. Marcel Dekker, New York (1993). (1993) Zbl0782.15001MR1200234
  4. Charkani, M. E., Boudine, B., 10.1007/s40863-020-00177-1, São Paulo J. Math. Sci. 14 (2020), 698-702. (2020) Zbl1451.13028MR4173484DOI10.1007/s40863-020-00177-1
  5. Chor, B., Rivest, R. L., 10.1109/18.21214, IEEE Trans. Inf. Theory 34 (1988), 901-909. (1988) Zbl0664.94011MR0982801DOI10.1109/18.21214
  6. Eisenbud, D., 10.1007/978-1-4612-5350-1, Graduate Texts in Mathematics 150. Springer, Berlin 1995. Zbl0819.13001MR1322960DOI10.1007/978-1-4612-5350-1
  7. Hachenberger, D., Jungnickel, D., 10.1007/978-3-030-60806-4_5, Topics in Galois Fields Algorithms and Computation in Mathematics 29. Springer, Cham (2020), 197-239. (2020) MR4233161DOI10.1007/978-3-030-60806-4_5
  8. Rabin, M. O., 10.1137/0209024, SIAM J. Comput. 9 (1980), 273-280. (1980) Zbl0461.12012MR0568814DOI10.1137/0209024
  9. Rotthaus, C., 10.1216/rmjm/1181071964, Rocky Mt. J. Math. 27 (1997), 317-334. (1997) Zbl0881.13009MR1453106DOI10.1216/rmjm/1181071964

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