# Unit or (Left/right zero) divisior [duplicate]

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?

• Hint: the mapping you wrote is injective if and only if it's surjective. – user228113 May 11 '15 at 21:32
• 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. – user228113 May 11 '15 at 21:35
• I added the "ring theory" tag; hope that's OK with you. Cheers! – Robert Lewis May 11 '15 at 22:02
• Please use the search feature before asking. – rschwieb May 11 '15 at 22:47