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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 <inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>. The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> as introduced in (Alahmadi et al, 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over <inline-formula> <tex-math notation="LaTeX">$\mathop {\mathrm {\mathbb {F}}} _{4}$ </tex-math></inline-formula>. We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>.