Is there a ring with $\mathbb{Z}$ as its group of units?
More generally, does anyone know of a sufficient condition for a group to be the group of units for some ring?
Is there a ring with $\mathbb{Z}$ as its group of units? More generally, does anyone know of a sufficient condition for a group to be the group of units for some ring? 


The example provided by Noam answers the first question. The second question is very old and, indeed, too general. See e.g. the notes to Chapter XVIII (page 324) of the book "László Fuchs: Pure and applied mathematics, Volume 2; Volume 36". In particular, rings with cyclic groups of units have been studied by RW Gilmer [Finite rings having a cyclic multiplicative group of units, Amer. J. Math 85 (1963), 447452], by K. E. Eldridge, I. Fischer [D.C.C. rings with a cyclic group of units, Duke Math. J. 34 (1967), 243248] and by KR Pearson and JE Schneider [J. Algebra 16 (1970) 243251]. 

