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I need help to solve this exercise. If anyone can help, thanks in advance!

Let $k$ a field and $R=k[x,y,z,t]/(xy-z^3,z^5,x^2t^3+y^2)$. Find the Krull dimension of $R$.

user26857
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user73454
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1 Answers1

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Let's find the height of the ideal $I=(xy-z^3,z^5,x^2t^3+y^2)$. By definition, this is the minimum height of the minimal prime ideals containing $I$. Let $P$ be a minimal prime containing $I$. Then $z\in P$, so $xy\in P$. If $x\in P$, then $y\in P$, and therefore $(x,y,z)\subseteq P$, hence equality. If $y\in P$, then $x\in P$ or $t\in P$, and thus $(x,y,z)\subseteq P$ or $(y,z,t)\subseteq P$, hence equality. In both cases we get that height of $P$ is three. It follows that the dimension of your ring is one.

user26857
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