# Where do non-associative rings appear?

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.)

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Feel free to make this a communication wiki. (I do not know how to do.) –  H. Shindoh Feb 20 at 17:08
One example would be the well-known Cayley-Dickson construction, which produces a sequence of increasingly "weird" algebras. For instance, complex numbers -> quaternions -> octonions (which are non-associative) -> .... It seems more of a curiosity than a serious theory, though. –  Marcin Łoś Feb 20 at 17:09
In the context of higher algebra, where you want things to be associative up to higher homotopy and so forth, you would say "strictly associative" instead of "associative." –  Qiaochu Yuan Feb 21 at 19:01

For example, $S^7$, the unit octonions, is a non-associative H-space under octonion multiplication. A famous result in algebraic topology asserts that $S^1, S^3, S^7$ (the unit complex numbers, the unit quaternions, and the unit octonions respectively) are the only spheres admitting H-space structures.