  In a finite abelian group $G$, define an additive matching to be a collection
of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i
= j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of
Croot-Lev-Pach and Ellenberg-Gijswijt, proved a polynomial upper bound on the
size of an additive matching. Fox and Lov\'{a}sz used this to deduce polynomial
bounds on Green's arithmetic removal lemma in $\mathbb{F}_2^n$.
  If $G$ is taken to be an arbitrary finite abelian group, the questions of
bounding the size of an additive matching and giving bounds for Green's
arithmetic removal lemma are much less well understood. In this note, we adapt
the methods of Fox and Lov\'{a}sz to prove that, provided we can assume a
sufficiently strong bound on the size of an additive matching in cyclic groups,
a similar bound should hold in the case of removal.
