What are some applications of noncommutative rings to questions which do not involve them in their statement? What are some external motivations and how does the known theory meet our hopes/expectations?
I’m aware of the Wedderburn theorem and its neat application to finite group representations, but off the top of my head that’s the only one I recall.
I guess technically Lie algebras count, but it seems they have their own neatly-packaged theory which is used all over the place. I prefer to exclude them from the question because of this distinct flavor, but would enjoy explanations of why this preference is misguided.