When reading papers on algebraic topology, I often find the term "associative ring".
The multiplication structure of a ring is normally assumed to be associative, therefore I guess that non-associative rings are important to some theories. But I have never seen such a theory.
Where can I find a non-associative ring? How important is it?
Edit: I'm very sorry for my misleading question. I am asking the reason why topologists need to mention "associative". A ring is associative by definition. So it is OK just to say "ring".
For example, "a Hausdorff manifold" is strange to say, because a manifold is assumed to be Hausdorff by definition. But some context, say Lie groupoids, "non-Hausdorff manifold" has meaning, so "Hausdorff manifold" is not strange. I am asking about such a context for "associative rings".
Maybe "an associative ring" appears in talks/papers relating to chain complexes (e.g. a paper/talk on model categories). So in some context, we need to deal with general rings whose multiplication needs not be associative for some reason. (Otherwise "associative ring" is still strange.)