Aliced, could you be a little more desciptive please? Are the 'rotten spots' mouldy or dry? Are they at the end of the fruit furthest from the stem? Are the 'rotten spots' healed over or are there holes in the skin?
There is a condition called "Blossom End Rot" which is a blackish looking dry spot or spots on the end of the fruit furthest from the stem. If this is what you have then there are one or two causes.
1. The plant is experiencing a lack of calcium due to deficiencies in the soil.
2. (The most common cause) The plant is receiving inconsistent moisture. Going from wet soil to dry soil and back again. Tomatoes like to have consistently moist (but not wet) soil. Going from wet to dry causes stress in the plant and the plant then has trouble absorbing calcium which is present in the soil. This causes a deficiency in the plant and leads to blossom end rot.
3. Too much nitrogen. An excess of nitrogen due to over fetilization causes the plant to grow and put out masses of leafy vegetation. The roots don't develop fast enough to support all the leaves AND the fruit at the same time. This causes nutrient deficiency (calcium) in the fruit leading to blossom end rot.
I hope this helps somewhat. If these aren't the type of "rotten spots" you have a problem with then a further description of the problem would help considerably.

Welcome, Aliced!
Are the spots on the bottom of the tomato, where the blossom used to be? If so, it is probably blossom end rot, which isn't a disease, but is a calcium deficiency in the fruit. It is primarily caused by a lack of water at night, when root pressure normally pushes calcium into the fruit. The first thing I would try is watering more deeply, and mulching to conserve water and even out soil moisture. If that doesn't work, and the soil is acid, add calcium in the form of lime.

Whoops, Gilli! We posted at the same time!