2
$\begingroup$

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

Thanks in advance.

$\endgroup$
4
$\begingroup$

An easy counterexample occurs with $A=\mathbb{Z}$, $p=2$, and $u_1=1$, $u_2=-1$.

$\endgroup$
3
  • $\begingroup$ Did exist a characterization of the fact " A have p units such that $u_1 + \dots + u_p =0$ for p prime ? $\endgroup$
    – user73577
    May 13 '13 at 6:07
  • 1
    $\begingroup$ in any unitary ring, $\;1+(-1)=0\,$ ... $\endgroup$
    – DonAntonio
    May 13 '13 at 6:10
  • $\begingroup$ I think $p$ is supposed to be given (not chosen) and the characterization is allowed to depend on $p$. $\endgroup$
    – Ted
    May 13 '13 at 6:19
4
$\begingroup$

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 $\mathbb{Z}/3 \mathbb{Z}.$

$\endgroup$
2
  • $\begingroup$ Did exist a characterization of the fact " A have p units such that $u_1 + \dots + u_p =0$ for p prime ? $\endgroup$
    – user73577
    May 13 '13 at 6:07
  • $\begingroup$ I don 't know if there is a characterization, but take any field of characteristic greater than p(p-1), where p is any prime. Then 1 + 2 + ..... + (p-1) + (-p(p-1)/2) is a zero sum of p distinct units. $\endgroup$ May 13 '13 at 22:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.