  Jespers and Sun conjectured that if a finite group $G$ has the property ND,
i.e. for any nilpotent element $n$ in the integral group ring $\mathbb{Z}G$ and
any primitive central idempotent $e \in \mathbb{Q}G$ one still has $ne \in
\mathbb{Z}G$, then at most one of the simple components of the group algebra
$\mathbb{Q} G$ has reduced degree bigger than $1$. With the exception of one
very special series of groups we are able to answer their conjecture, showing
that it is true - up to exactly one exception. To do so we first describe
groups with the so-called SN property which was introduced by Liu and Passman
in their investigation of the Multiplicative Jordan Decomposition for integral
group rings.
  We then study further objects connected to the property ND. This concerns on
one hand a certain section of the unit group of $\Z G$ which measures how far
$G$ is from having ND and about which Jespers and Sun posed two further
questions which we answer. On the other hand we introduce two properties which
appeared naturally in these investigations: one is purely
representation-theoretic, while the other can be regarded as indicating that it
might be hard to decide ND. Among others we show these two notions are
equivalent for groups with SN.
