The cores usually are not *that* radioactive. The heat primarily comes from the time they were formed, and they contiunously lose warmth to the surrounding space.
Now, volcanic activity can happen due to a lot of causes. It can start through tectonic activity, gravitational influence, meteor impact, planet climate conditions, etc.
Now for mars it (probably) has the following reasons:
* Less own gravity than earth - lava can more easily be transported to the planet's surface
* Mars volcanoes are fix spots since the surface doesn't move (no tectonic activity)
* The volcanic activity comes from Mars' rotational axis wobbling around
* The crust in the southern hemisphere is thicker - more rock melts on the north side and the landscape sinks till an isostatic balance is reached. This causes gravitational anomalies and thus - volcanism.