Research Article
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Year 2017, , 115 - 122, 09.01.2017
https://doi.org/10.13069/jacodesmath.284939

Abstract

References

  • [1] V. Camillo, W. K. Nicholson, On rings where left principal ideals are left principal annihilator, Int. Electron. J. Algebra 17 (2015) 199–214.
  • [2] T. J. Dorsey, Morphic and principal–ideal group rings, J. Algebra 318(1) (2007) 393–411.
  • [3] J. L. Fisher, S. K. Sehgal, Principal ideal group rings, Comm. Algebra 4(4) (1976) 319–325.
  • [4] P. Hurley, T. Hurley, Module codes in group rings, Proc. Int. Symp. Information Theory (ISIT) (2007) 1981–1985.
  • [5] P. Hurley, T. Hurley, Codes from zero–divisors and units in group rings, Int. J. Inf. Coding Theory (2009) 57–87.
  • [6] S. Jitman, S. Ling, H. Liu, X. Xie, Checkable codes from group rings, arXiv:1012.5498, 2010.
  • [7] F. J. MacWilliams, Codes and ideals in group algebras, Combinatorial Mathematics and its Applications (1969) 317–328.
  • [8] W. K. Nicholson, E. Sánchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra 271(1) (2004) 391–406.

Code–checkable group rings

Year 2017, , 115 - 122, 09.01.2017
https://doi.org/10.13069/jacodesmath.284939

Abstract

A code over a group ring is defined to be a submodule of that group ring. For a code $C$ over a group ring $RG$, $C$ is said to be checkable if there is $v\in RG$ such that {$C=\{x\in RG: xv=0\}$}. In \cite{r2}, Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring $RG$ is code-checkable if every ideal in $RG$ is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring $\mathbb{F}G$, when $\mathbb{F}$ is a finite field and $G$ is a finite abelian group, to be code-checkable. In this paper, we give some characterizations for code-checkable group rings for more general alphabet. For instance, a finite commutative group ring $RG$, with $R$ is semisimple, is code-checkable if and only if $G$ is $\pi'$-by-cyclic $\pi$; where $\pi$ is the set of noninvertible primes in $R$. Also, under suitable conditions, $RG$ turns out to be code-checkable if and only if it is pseudo-morphic.

References

  • [1] V. Camillo, W. K. Nicholson, On rings where left principal ideals are left principal annihilator, Int. Electron. J. Algebra 17 (2015) 199–214.
  • [2] T. J. Dorsey, Morphic and principal–ideal group rings, J. Algebra 318(1) (2007) 393–411.
  • [3] J. L. Fisher, S. K. Sehgal, Principal ideal group rings, Comm. Algebra 4(4) (1976) 319–325.
  • [4] P. Hurley, T. Hurley, Module codes in group rings, Proc. Int. Symp. Information Theory (ISIT) (2007) 1981–1985.
  • [5] P. Hurley, T. Hurley, Codes from zero–divisors and units in group rings, Int. J. Inf. Coding Theory (2009) 57–87.
  • [6] S. Jitman, S. Ling, H. Liu, X. Xie, Checkable codes from group rings, arXiv:1012.5498, 2010.
  • [7] F. J. MacWilliams, Codes and ideals in group algebras, Combinatorial Mathematics and its Applications (1969) 317–328.
  • [8] W. K. Nicholson, E. Sánchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra 271(1) (2004) 391–406.
There are 8 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Noha Abdelghany This is me

Nefertiti Megahed This is me

Publication Date January 9, 2017
Published in Issue Year 2017

Cite

APA Abdelghany, N., & Megahed, N. (2017). Code–checkable group rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 115-122. https://doi.org/10.13069/jacodesmath.284939
AMA Abdelghany N, Megahed N. Code–checkable group rings. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2017;4(2 (Special Issue: Noncommutative rings and their applications):115-122. doi:10.13069/jacodesmath.284939
Chicago Abdelghany, Noha, and Nefertiti Megahed. “Code–checkable Group Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 2 (Special Issue: Noncommutative rings and their applications) (May 2017): 115-22. https://doi.org/10.13069/jacodesmath.284939.
EndNote Abdelghany N, Megahed N (May 1, 2017) Code–checkable group rings. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 115–122.
IEEE N. Abdelghany and N. Megahed, “Code–checkable group rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), pp. 115–122, 2017, doi: 10.13069/jacodesmath.284939.
ISNAD Abdelghany, Noha - Megahed, Nefertiti. “Code–checkable Group Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (May 2017), 115-122. https://doi.org/10.13069/jacodesmath.284939.
JAMA Abdelghany N, Megahed N. Code–checkable group rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:115–122.
MLA Abdelghany, Noha and Nefertiti Megahed. “Code–checkable Group Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), 2017, pp. 115-22, doi:10.13069/jacodesmath.284939.
Vancouver Abdelghany N, Megahed N. Code–checkable group rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):115-22.