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We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>=</mo><mfenced separators="" open="⟨" close="⟩"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><mn>2</mn><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace></mrow><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mi>b</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi mathvariant="italic">ab</mi><mo>=</mo><mn>0</mn></mfenced></mrow></semantics></math></inline-formula> and the noncommutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>=</mo><mfenced separators="" open="⟨" close="⟩"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><mn>2</mn><mi>a</mi><mo>=</mo><mn>2</mn><mi>b</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em"></mspace></mrow><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mi>a</mi><mo>,</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mi>b</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi mathvariant="italic">ab</mi><mo>=</mo><mi>a</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi mathvariant="italic">ba</mi><mo>=</mo><mi>b</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.