Zero divisors in $A[x_1,x_2,\dots,x_r]$

Let $A$ be a commutative ring. I am trying to show that if $f(x_1,x_2,\ldots, x_r) \in A[x_1,x_2,\ldots, x_r]$ is a zero divisor then there exists $a$ in $A-\{0\}$ such that $af=0$ in $A[x_1,x_2,\ldots, x_r]$.

What I have tried so far as following.
I am using induction on $r$ (not fixing the ring). The base case is for $r=1$ which I did assuming the minimal degree of $g$ for which $fg=0$ holds. So my induction assumption is for $r \geq 2$ whenever $f$ is in $A[x_1,x_2,\ldots x_r]$, $n<r$ is a zero divisor there is $a$ in $A-\{0\}$ such that $af=0$. So for the final step I take $f(x_1,x_2,\ldots, x_r) \in A[x_1,x_2,\ldots, x_r]$ is a zero divisor. Now since $A[x_1,x_2,\ldots, x_r]=A[x_1][x_2,x_3,\ldots, x_r]$ by induction hypothesis there is $g$ in $A[x_1]-\{0\}$ such that $fg=0$.

I cannot proceed further. Am I correct so far? Please help me. Thank you.

Solution 1:

From your reasoning it follows that $g(x_1)$ annihilates the (non-zero) coefficients of $f$ in $A[x_1]$. All you have to do now is to prove the following extension of the case $r=1$:

Let $f_1,\dots,f_t\in A[X]$ be non-zero polynomials. If there is $g\in A[X]$, $g\ne 0$, such that $gf_1=\cdots=gf_t=0$, then there is $a\in A$, $a\ne 0$, such that $af_1=\cdots=af_t=0$.

We show by induction on $d=\max_{1\le i\le t}\deg f_i$ that there is an element $a$ as before which moreover belongs to the ideal generated by the coefficients of $g$.

If $d=0$ there is nothing to prove. For $d\ge 1$ let's write the polynomials $f_i(X)=\sum_{j=0}^{n_i}a_{ij}X^j$ with $n_i=\deg f_i$, $\max\{n_1,\dots,n_t\}=d$, and $g(x)=\sum_{j=0}^nb_jX^j$, where $n=\deg g$. Suppose that $n\ge 1$. Then $b_na_{in_i}=0$ for $i=1,\dots,t$. Now consider two cases:
(1) If $a_{1n_1}g=\cdots=a_{tn_t}g=0$, then set $f_i'=f_i-a_{in_i}X^{n_i}$. Since $gf_i'=0$ for $i=1,\dots,t$ and $\max_{1\le i\le t}\deg f_i'<d$ we can apply the induction hypothesis and find an element $a\ne 0$ in the ideal generated by the coefficients of $g$ such that $af_1'=\cdots=af_t'=0$. This leads to $af_1=\cdots=af_t=0$.
(2) If there is $1\le i\le t$ such that $a_{in_i}g\ne 0$, then $(a_{in_i}g)f_1=\cdots=(a_{in_i}g)f_t=0$ and $\deg (a_{in_i}g)<\deg g$. An induction argument on $\deg g$ shows that there is $a\ne 0$ in the ideal generated by the coefficients of $a_{in_i}g$ such that $af_1=\cdots=af_t=0$.