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We prove the Liv\v{s}ic Theorem for H\"{o}lder continuous cocycles withvalues in Banach rings. We consider a transitive homeomorphism ${\sigma:X\toX}$ that satisfies the Anosov Closing Lemma, and a H\"{o}lder continuous map${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertibleelements of some Banach ring $B$. We show that it is a coboundary with aH\"{o}lder continuous transition function if and only if${a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)=e}$ for each periodic point$p=\sigma^n p$.

Liv\v{s}ic Theorem for Banach Rings