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I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping.

$$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ $$H^\prime=k+kt^{4\nu}(1+t+t^2)+kt^{6\nu}(1+t+t^2)+kt^{7\nu}(1+t+t^2)+k[[t]]t^{8\nu}$$

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I'm sorry but I don't even understand the notation (I'm sure it's standard). Is $k$ a field and does $+$ denote direct sum? –  Rudy the Reindeer May 24 at 14:06
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I understand that $k[[t]]$ is probably the ring of formal power series but what is $k[[t]]t^{8 \nu}$? –  Rudy the Reindeer May 24 at 14:07
    
In fact, This ring is called as an Arf ring @user26857 –  B11b May 25 at 17:41

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