Let R is finite Ring with 1 and $a \in R \setminus \{0\}$. Show that a is Unit or (Left/right) zero divisior It's obvious that we have to use the mapping: $x \rightarrow ax$ and $x \rightarrow xa$ but what else?

  • $\begingroup$ Hint: the mapping you wrote is injective if and only if it's surjective. $\endgroup$ – user228113 May 11 '15 at 21:32
  • $\begingroup$ Also: be careful that you need an extra argument to show that $a$ has a left-side inverse if and only if it has a right-side inverse, in the noncommutative case. $\endgroup$ – user228113 May 11 '15 at 21:35
  • $\begingroup$ I added the "ring theory" tag; hope that's OK with you. Cheers! $\endgroup$ – Robert Lewis May 11 '15 at 22:02
  • $\begingroup$ Please use the search feature before asking. $\endgroup$ – rschwieb May 11 '15 at 22:47