Your plant is called Yucca cane. The lower cane probably died because of a problem with the roots, most likely improper watering. In general, it needs a thorough watering as soon as the top inch of soil feels dry. Make sure it does not sit in water that collects in the bottom of the outer pot.
If there are no signs of healthy buds emerging from the shorter cane or if the shorter cane feels soft, then you can safely assume it will not revive and it can be removed.