  Kaplansky's zero divisor conjecture (unit conjecture, respectively) states
that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring
$\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater
than $1$). In this paper, we study possible zero divisors and units in
$\mathbb{F}[G]$ whose supports have size $3$. For any field $\mathbb{F}$ and
all torsion-free groups $G$, we prove that if $\alpha \beta=0$ for some
non-zero $\alpha, \beta \in \mathbb{F}[G]$ such that $|supp(\alpha)|=3$, then
$|supp(\beta)|\geq 10$. If $\mathbb{F}=\mathbb{F}_2$ is the field with 2
elements, the latter result can be improved so that $|supp(\beta)|\geq 20$.
This improves a result in [J. Group Theory, 16 (2013), no. 5, 667-693].
Concerning the unit conjecture, we prove that if $\alpha \beta=1$ for some
$\alpha, \beta \in \mathbb{F}[G]$ such that $|supp(\alpha)|=3$, then
$|supp(\beta)|\geq 9$. The latter improves a part of a result in [Exp. Math.,
24 (2015), 326-338] to arbitrary fields.
