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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

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Algebraic topologists like to study H-spaces, which are spaces equipped with a multiplication map which is unital but need not be associative. Even this condition is already enough to impose strong constraints on the space. In particular, the homology of the space acquires a product, the Pontryagin product, which is again not necessarily associative.

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.

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One important class of examples would be Lie algebras, which arise as the tangent spaces of Lie groups. These are extremely important, in mathematics and in physics. The octonions are another non-associative ring with connections to geometry (and Lie groups).

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