This question is based on two answers I saw. I want to know if my reasoning is correct in each case. For me, $A$ is always a commutative ring with $1 \ne 0$.
In the first answer this proposition was given:
Let $f_1,\dots,f_t\in A[X]$ be nonzero. If there is $0 \ne g\in A[X]$ such that $gf_1=\cdots=gf_t=0$, then there is $0 \ne a\in A$ such that $af_1=\cdots=af_t=0$.
Assuming the case $t = 1$ which is McCoy's theorem, I think I can prove this as follows:
Proof: Let $\displaystyle F := \sum_{i=1}^t f_i X^{\sum_{j < i} (1 + \deg f_j)}$. Then $gf_1 = \ldots = gf_t = 0 \implies gF = 0 \implies aF = 0$ for some $0 \ne a \in A \implies af_1 = \ldots = af_t = 0$. The last implication is because every coefficient of each $f_i$ appears exactly once as a coefficient of $F$.$\newcommand{\N}{\mathbb{N}}$
Question 1: Is the proof above correct?
Moving on to the next question, in the other answer this proposition was given:
(*) Let $A$ be a commutative $\N$-graded ring. Let $f \in A$ be a zerodivisor. Then there is some $0 \ne a \in A$ homogeneous such that $af = 0$.
I think the proof given in that answer has a fatal flaw, which was pointed out as a comment but with no response. Specifically $\deg f_ig < \deg g$ may not be true. For polynomial rings in $1$ variable it's ok because $f_i = a_ix^i$ where $a_i$ has degree $0$ and $x^i$ is a nonzerodivisor, so $\deg a_ig < \deg g$ and $a_ig \ne 0$. So my second question is:
Question 2: Is (*) true? If not, what is a counterexample?
One note is that (*) is true if $A$ is Noetherian (since then every zerodivisor is in some associated prime, which is the annihilator of a homogeneous element). So a counterexample would have to be non-Noetherian.
