Yes after opening the Red Wine you should put the cork back in the Red Wine bottle and then place it in the refrigerator to keep it from spoiling.
The Red Wine when corked in the refrigerator should last up to 5 days without going bad so as long as you drink the Red Wine within 5 days the Wine should be good to drink.
After 5 days the Wine should be thrown out because the spoiled Wine could make you sick.