A finer classification of the unit sum number of the ring of integers of quadratic fields and complex cubic fields

The unit sum number, u(R), of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending on whether the units generate R additively or not. Here we introduce a finer classification for the unit sum number of a ring and in this new classification we completely determine the unit sum number of the ring of integers of a quadratic field. Further we obtain some results on cubic complex fields which one can decide whether the unit sum number is ω or ∞. Then we present some examples showing that all possibilities can occur.