ON DUAL SURJUNCTIVITY AND APPLICATIONS

The paper establishes Theorem 4.1 correctly: for any surjunctive or dual surjunctive group G and any field K of positive characteristic, K[G] is directly finite. The key operator identity is Tb ∘ Ta = Tab, so from ab = 1 one gets Tb ∘ Ta = Id, implying Ta is injective and Tb is post‑surjective; surjunctivity or dual surjunctivity then forces bijectivity and hence ba = 1 (finite-field case), and a standard reduction argument extends to all positive characteristic fields . The candidate solution incorrectly uses the composition rule Tc ∘ Td = Tcd (the correct rule is Tc ∘ Td = Tdc), which flips injectivity/surjectivity roles and leads to a misapplication of post‑surjectivity; its general-field lift via Jacobson rings would be fine only if the prior step were correct, which it isn’t .