On $\Sigma$-nilpotent ideals of topological PI-rings

We show that under certain conditions on the topology of a faithful module $M$ over a topological PI-ring $R$, if $M$ has at most countable dual topological Krull dimension, then the closure of the sum of all $\Sigma$-nilpotent ideals of the ring $R$ is a $\Sigma$-nilpotent ideal too, and in the case of a bounded ring $R$ its topological Baer radical is $\Sigma$-nilpotent.