This is a problem from a test I took today.
Definition: A pretty ring $R$ is a ring with unity 1, not a field, and each nonzero element can be written uniquely as a sum of a unit and a nonunit element of $R$.
The first problem, which is much easier, is to find such a ring. A got difficulty at first but came out with $$R = \left\{\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\mid a,b \in \mathbb{Z}_2\right\}$$ with usual operation done in modulo $2$.
The latter problem, which I can't solve yet is to find all possible characteristic of pretty ring.
From some experiment with almost the same ring, I came to conclusion that a pretty ring can only has characteristic 2. But, I can't prove it for the general case.
We show that a pretty ring has exactly one unit. Indeed, if $0 \neq u$ is not a unit and $e$ is a unit, then $$ (u+e) + 0 = e + u $$ tells us that $u+e$ is not a unit (otherwise we get the contradiction $u=0$). Nowe take two units $e, \tilde{e}$ then $$ e + (u+ \tilde{e}) = \tilde{e} + (u+ e) $$ implies $e=\tilde{e}$ and hence $1$ is the only unit.
On the other hand every ring with only one unit is a pretty ring as we can write every $x\neq 0$ as $$ x = 1 + (x-1).$$
Thus, we have for a unital ring $R\neq \mathbb{Z}/2 \mathbb{Z}$: $$ R \text{ is a pretty ring} \quad \Leftrightarrow \quad \vert R^\times \vert =1 $$ In particular we have $-1=1$ and thus a pretty ring has either characteristic equal to $1$ or $2$. Note that both cases are possible as the zero ring is a pretty ring.
Shorter proof: Assume that that the characteristic of the pretty ring is not $2$. Then we get from $1+0=-1+2$ must be a unit. Let $u\neq 0$ be a non-unit (exists as a pretty ring is not a field), then $1+0=(u+1) - u$ implies that $u+1$ is a non-unit. Then we get from $(u+1)+1=u +2$ that $0=1$, ie. our pretty ring is the zero ring. Therefore, a pretty ring has characteristic $1$ or $2$.