Cyclic Codes over a Commutative Non-Unital Ring

This paper investigates cyclic codes over the commutative non-unital ring $I=⟨a,b\mid 2a=2b=0,a^2=b,ab=0⟩$, a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over I, analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from $I^n$ to ${\mathbb{F}}_2^{2n}$ is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths $2\le n\le 7$ illustrate the diversity of cyclic codes over I and highlight their potential applications.