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 ɛ rg such that c = {x ɛ 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.