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The Quarterly Journal of Mathematics 2005 56(1):1-12; doi:10.1093/qmath/hah023
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© The Author (2005). Published by Oxford University Press. All rights reserved. For Permissions please email: journals.permissions@oupjournals.org

On the unit sum number of some rings

Nahid Ashrafi1 * and Peter Vámos2 §

1 Mathematics Department, University of Semnan, Semnan, Iran, 2 School of Mathematical Sciences, University of Exeter, Exeter, EX4 4QE

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 {omega} or {infty} 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) ≥ {omega}, 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 {omega} when neither is algebraic and one of them is pure transcendental.

Received 21 July 2003. Revised 5 May 2004.

* E-mail: nashrafi{at}semnan.ac.ir

§ Corresponding author. E-mail: P.Vamos{at}ex.ac.uk


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