  We show that in a tracial and finitely generated $W^\ast$-probability space
existence of conjugate variables in an appropriate sense exclude algebraic
relations for the generators. Moreover, under the assumption of finite
non-microstates free Fisher information, 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.
