Is there any trivial ring which isn't null? By a trivial ring, I mean one that fulfills the following: ${\forall}x,y\,{\in}\,R:xy=0_R$
A null ring is a ring with only one element.
So far I couldn't think of any trivial ring which isn't null. Any examples?

$(\{0,a\},+,\cdot)$ with $a+a:=0$ and $a\cdot a:=0$ is a rng, i.e. a ring without $1$. It is structurally equivalent to the $\{[0],[2]\}$ subring of $\mathbb Z_4$ user307935 suggested.
If, on the other hand, you require an $1$ element then you can satisfy $1\cdot 1=1$ (from the ring axioms) and $1\cdot 1=0$ (from your constraint) only by $1=0$, so you'd be back to the null ring.
