Do Pineapples Ripen After They Are Picked
Pineapple softens, but not sweeten on shelf ripening. 

Fruit ripening is usually due to exposure of unsaturated hydrocarbons e.g. ethylene. This phenomenon occurs in fruit known as "climacteric fruits". Well known examples are banana, avocado, tomato, apple, pear, kiwi, to name a few.

Pineapple is not a kind of climacteric fruit. However, ethylene exposure will activate chitinase (which is a type of enzyme that breaks down chitin, a structural material inside plants), and cause softening of the pineapple. To achieve this a simple method is to place a ripe banana with the pineapple which release ethylene.