Zero divisors in a Ring Let's say $a, b$ are zero divisors in a Ring, (i.e., there exist some $x,y \in R$ s.t. $ax=0, by=0$).
I feel that $a$ is a zero divisor of $xy$ (as $axy=0y=0$), but is $b$ a zero divisor of $xy$?
If I take a look at $bxy$, I know I can't commute $b$ and $x$, but can $b$ be a zero divisor of $(xy)$?

Your intuition is right. Indeed, $axy=0$, but we cannot say that $bxy=0$.
For example, consider $M_2(\mathbb{Z})$, the $2\times2$ matrices with integer coefficients. We have that
\begin{align*}
\begin{pmatrix}
1&0\\0&0
\end{pmatrix}
\begin{pmatrix}
0&0\\1&1
\end{pmatrix}=\begin{pmatrix}
0&0\\0&0
\end{pmatrix}
\end{align*}
and
\begin{align*}
\begin{pmatrix}
0&1\\0&0
\end{pmatrix}
\begin{pmatrix}
1&1\\0&0
\end{pmatrix}=\begin{pmatrix}
0&0\\0&0
\end{pmatrix}
\end{align*}
It is clear that
\begin{align*}
\begin{pmatrix}
1&0\\0&0
\end{pmatrix}
\begin{pmatrix}
0&0\\1&1
\end{pmatrix}
\begin{pmatrix}
1&1\\0&0
\end{pmatrix}=\begin{pmatrix}
0&0\\0&0
\end{pmatrix}
\end{align*}
but
\begin{align*}
\begin{pmatrix}
0&1\\0&0
\end{pmatrix}
\begin{pmatrix}
0&0\\1&1
\end{pmatrix}
\begin{pmatrix}
1&1\\0&0
\end{pmatrix}=\begin{pmatrix}
1&1\\0&0
\end{pmatrix}
\end{align*}
