Let $R$ be a ring and $S$ be an ideal of $R$.
Let $Ann_R(S)_r=\{a\in R | Sa=0\}$ and $Ann_R(S)_l=\{a\in R | aS=0\}$ denote the right annihilator and the left annihilator of $S$ in $R$, respectively.
If $R$ is a semiprime ring (i.e Given an ideal $S$, a semiprime ring is one for which $S^n=0$ implies $S=0$ for any positive $n$), then $Ann_R(S)_r=Ann_R(S)_l$.
$A=Ann_R(S)_rS$ is an ideal.
$A^2=\{0\}$.
If $R$ is semiprime, this means $A=\{0\}$.
This means $Ann_R(S)_r\subseteq Ann_R(S)_l$.
By symmetry, the reverse containment holds.
QED.
One can also compare this to the obvious element-wise version that holds for noncommutative reduced rings: if $xy=0$ then $yx=0$. Proof: $(yx)^2=0$, so $yx=0$.