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Let $R=k[t]/(t^2)$ and $S=k[t,x]/(t^2,tx^3+tx^2-x^2-x)$, $k$ is a field. I must prove that $S$ is integral over $R$ and that $S=R\oplus R$.

Any help about that..thanks in advance...

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georgeson1960
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1 Answers1

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Notice that $S = R[x]/((tx-1)(x^2+x))$. Since $t$ is nilpotent in $R$, $tx-1$ is a unit, so in fact $S = R[x]/(x^2+x) = R[x]/(x(x+1)) \cong R[x]/(x) \oplus R[x]/(x+1)$, by Chinese Remainder (since $(x), (x+1)$ are comaximal). But $R[x]/(x) \cong R \cong R[x]/(x+1)$, so this shows $S \cong R \oplus R$. Thus $S$ is a finite $R$-module, so the extension $R \subseteq S$ is integral (alternatively, $x \in S$ satisfies the monic polynomial $f(z) = z^2 + z$ over $R$).

zcn
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