Abstract Zero-divisor codes are constructed using group rings where their generators zero-divisors. Generally, zero-divisor can be equivalent despite associated groups non-isomorphic, leading to the proposed conjecture “Every dihedral code has an form of cyclic code”. This paper is devoted study equivalence in $$F_2G$$ <mml:ms...

Reopening a cold case, Inspector Echelon, high-ranking in the Row Operations Center, is searching for lost linear map, known to be nilpotent. When partially decomposed matrix unearthed, he reconstructs its reduced form, finding it singular. But were origins nilpotent?