On the unit sum number of some rings

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 whether the units generate R additively or not. We show that if R is the ring of integers of a quadratic or complex cubic number field then u(R) ≥ ω, in the quadratic case u(R) is completely determined. We also investigate tensor products of algebras. If the ground field is not the field of two elements then their unit sum number is 2 when at least one of them is algebraic and at least ω when neither is algebraic and one of them is pure transcendental.