A ring is a set $S$ together with two operations: $*$ and $+$. The operations in question have the following properties: $*$ distributes over $+$, $(R,+)$ forms an Abelian group and $S$ is closed under both operations. That's for sure. There's also associativity of $*$ - some sources I've used list it as another axiom for rings, and I have thought this was the case, but I've just encountered a notion of "non-associative ring" - apparently, such objects exist and there are theorems about them.
Why are rings often defined to be associative, if mathematicians work on non-associative objects which are too called "rings"?