  We show that in a tracial and finitely generated $W^\ast$-probability space
existence of conjugate variables excludes algebraic relations for the
generators. Moreover, under the assumption of maximal non-microstates free
entropy dimension, we prove that there are no zero divisors in the sense that
the product of any non-commutative polynomial in the generators with any
element from the von Neumann algebra is zero if and only if at least one of
those factors is zero. In particular, this shows that in this case the
distribution of any non-constant self-adjoint non-commutative polynomial in the
generators does not have atoms.
  Questions on the absence of atoms for polynomials in non-commuting random
variables (or for polynomials in random matrices) have been an open problem for
quite a while. We solve this general problem by showing that maximality of free
entropy dimension excludes atoms.
