Wobbly bottom bracket on Radford, need guidance to replace
What you have in the picture is a cartridge-style bottom bracket which contains the bottom bracket bearings. What you called the 'pedal stems' are the cranks.

Cranks should be removed from the square tapers on the bracket shaft with a special crank puller tool such as the Park Tool CWP-7. If you forced the cranks off by other means you have possibly damaged them. If re-installed on a bottom bracket they may keep working loose.

The bottom bracket is removed using a bottom bracket tool such as the Park Tool BBT-22 that fits into the splines in the cartridge body.