# Ring with Z as its group of units?

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?

• $k[X,1/X]$ where $k$ is the 2-element field. Sep 12 '11 at 3:53
• A necessary condition is of course that $-1 = 1$ Sep 12 '11 at 5:53
• The group $G$ is always contained in the group of units of the group ring $R[G]$, when $R$ is a commutative ring with unit. I don't know precise conditions for when they are equal, but here's a reference: maths.ed.ac.uk/~aar/papers/higman.pdf Sep 12 '11 at 6:58
• I was thinking also to look at the group ring $\mathbb{F}_2(G)$; but the units of this ring are strictly more than just the elements of $G$, for example, if $G$ is a finite $2$-group of order greater than $2$. See the second paragraph of www.ieja.net/papers/2011/V9/13-V9-2011.pdf Sep 12 '11 at 8:30
• Since the last part of the question has been asked again in a slightly different way, I thought I would add a comment that an example of a group which is not the group of units of any ring is the cyclic group of order 5. This is a nice exercise. Oct 19 '11 at 8:23