Let $k$ be a field, and let $G$ be a countable nilpotent group with centre $Z$. We show that the group algebra $kG$ is primitive if and only if $k$ is countable, $G$ is torsion free, and there exists an abelian subgroup $A$ of $G$, of infinite rank, with $A \cap Z = 1$. Suppose now that $G$ is torsion free. Then $kG$ has a partial quotient ring $Q = kG{(kZ)^{ - 1}}$. The above characterisation of the primitivity of $kG$ is intimately connected with the question: When is $Q$ a Noetherian ring? We determine this for those groups $G$, as above, all of whose finite rank subgroups are finitely generated. In this case, $Q$ is Noetherian if and only if $G$ has no abelian subgroup $A$ of infinite rank with $A \cap Z = 1$.