In this paper we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely I, E, and H as denoted in (Fine, 1993). A new notion of duality is introduced in the case of E that coincides with the notion of quasi self-dual code introduced in (Alahmadi et al, 2022). We characterize selfdual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over F 4. We study the connection between the dual of any linear code over these rings and the dual of its residue and torsion codes. A MacWilliams formula is established for linear codes over the non-commutative ring E.