# Units on a ring

Let $A$ be a ring such that there exists units $u_1, \dots , u_p : u_1 + \dots + u_p=0$ and $p$ a prime. Did $A$ have a subring isomorphic to $\mathbb{Z}/p \mathbb{Z}$ in general

An easy counterexample occurs with $A=\mathbb{Z}$, $p=2$, and $u_1=1$, $u_2=-1$.
• Did exist a characterization of the fact " A have p units such that $u_1 + \dots + u_p =0$ for p prime ? May 13 '13 at 6:07
• in any unitary ring, $\;1+(-1)=0\,$ ... May 13 '13 at 6:10
• I think $p$ is supposed to be given (not chosen) and the characterization is allowed to depend on $p$.
The answer is clearly no in general. The $3$ non-zero elements of a field of order $4$ have sum zero, but there is no subring $\mathbb{Z}/3 \mathbb{Z}.$
• Did exist a characterization of the fact " A have p units such that $u_1 + \dots + u_p =0$ for p prime ? May 13 '13 at 6:07