Code–checkable group rings
Code–checkable group rings
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.
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- [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.