Code–checkable group rings

Abstract

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 ∈ RG such that C = {x ∈ RG : xv = 0}. In [6], 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 FG, when 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 π -by-cyclic π; where π 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.