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In exercise 1.8 of chap I in Hartshorne algebraic geometry,

Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y \nsubseteq H$. Then every irreducible component of $Y \cap H$ has dimension $r-1$.

Let $I(H)={f}$. I saw a solution in which one of the steps was to prove that $\bar{f}$ is not a unit in $A/I(Y)$. How to prove this ?

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    I guess if $f$ were a unit in $A/I(Y)$, then $Y \cap H$ would be empty and the problem is trivially true. – 2017-02-02
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    @hwong557 you should submit that as an answer. – 2017-02-02

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