First: the 3 "cylinders" on the base start from a unique base? Or have separate roots?
If they have separate roots =
Second: grab all 3 cylindres from the ground. Throw off the whole rotten cylinder. Stretch the other two (separately) on a bed of dry sand or newsprint. Leave the two cylinders to rest for 7-8 days. Look carefully if you see traces of rot in the two. If you see any rot, choose the cylinder healthy. Or, if both are healty, follow so.
3th: Throw off the old pot and all the soil. Use a new pot of clay (which allows the perspiration) for each healty cylinder, separately, no plastic and no glazed ceramic. Check that the pots have drainage holes. Put a piece of broken pot (not the old one) over the drainage hole. Put little gravel at the bottom for better drainage. Fill the pot with sand e replants separately the two cylinders.
Don't water them in this phase.
After one month, take the cylinders from the sand and check if you see any rot.
If not, change the soil into the pot (good soil for cacti is: 1/3 clay, 1/3 sand, 1/3 ground of the forest) and replant your 2 cuttings. DON'T WATER IT for 7-10 days. After that you can go on with normal watering.
Count that if you got a rot, that means that almost surely your cactus had too much water with respect to the light. So look for a place with more sun, or give less water.
Follow the Bamboo's suggests for watering.
If some (or both) of your cylinders shows rot. You can save it.
Cut the cylinder at least 30 cm over the rot. To cut use a saw well disinfected with hydrogen peroxide. Re-disinfect it if you have 2 cylinders, before the second cut. And protect your hands with hard gloves also.
Check the center of cylinders. If it is black or brown, cut again 30 cm over it, until you reach the completely healthy part.
Sprinkle the cut with antibiotic and healing powder for humans (Here is called Cicatrene). Stretch the two (separately) on a bed of dry sand or newsprint. Leave them to dry for 3-4 days (according to the diameter), always checking for rots. As soon as it is dry, cover all the cut with wax for grafting. Put the cylinder in a jar with sand only, until it gives the roots. You must also wet, in moderation.
After about a month, the roots should be grown (just a bit is enough, but treat it with carefully). Take the cylinder from sand, check the roots and eventually rot, and transplant it in its definitive pot. Follow the explanations above for sun and watering.
This operation is very delicate. Both because the rot go up in the drum together with the sap and a bit of bacteria can be already risen without you seeing it. And because it is very difficult to estimate the timings of transplant and how watering.
I hope, however, that my directions you might find useful.