  For which groups $G$ is it true that for all fields $k$, every non-monomial
element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The
only groups for which we know this are the torsion-free abelian groups. We
would like to know whether it also holds for all free groups.
  It is shown that the above property fails for wide classes of groups: for
every group $G$ that contains an element $g\neq 1$ whose image in $G/[g,G]$ has
finite order (in particular, every group containing a $g\neq 1$ that itself has
finite order, or that satisfies $g\in [g,G])$; and for every group containing
an element $g$ which commutes with a conjugate $hgh^{-1}\neq g$ (in particular,
for every nonabelian solvable group).
  Results are obtained on closure properties of the class of groups satisfying
the stated condition. Many further questions are raised; in particular, a
plausible Freiheitssatz for group algebras of free groups is noted.
