Non-Associative Rings

I recently did a course on Associative Rings. Do there exist non-Associative rings? I’d like to discuss certain examples here.

I’m told that $\Bbb{R}^3$ is one such ring, which is non-associative under the operation of vector product. In other words, $A\times (B\times C)\neq (A\times B)\times C$.

After thinking about it for a bit, I found a supporting example in no time. We can see that $i\times (i\times j)=-j$. However, $(i\times i)\times j=0$.

Wikipedia says that Lie Algebras are also examples of non-Associative rings. But that is a topic for another time.