Find an infinity ring with finite characteristic p, which has following properties:
(i) is not commutative
(ii) is a field
(iii) does not have unit
I think that the (ii) can be field of fractions using integral domain ${\displaystyle \mathbb {Z} }_p[x]$ set of all polynomials over the field $ {\displaystyle \mathbb {Z} }_p$
But I cannot find other examples. Probbaly some sequences with finite non-zero elements can work but I have no idea how to finish it.
Thank you very much for any help.
For (i), take the ring of $n\times n$ matrices for some $n>1$, with entries in $\Bbb Z_p[x]$. For (iii), take $p\Bbb Z_{p^2}[x]$.