| 1. | Every division ring is therefore a division algebra over its center.
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| 2. | The center of a division ring is commutative and therefore a field.
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| 3. | In fact a division ring is also a simple ring.
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| 4. | Consequently, the endomorphism ring of any simple module is a division ring.
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| 5. | The projective line over a division ring results in a single auxiliary point.
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| 6. | Unlike quaternions they do not form a division ring.
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| 7. | This shows that integral domains and division rings don't have such idempotents.
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| 8. | In particular, the endomorphism ring of a simple module is a division ring.
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| 9. | Moufang planes are coordinatized by alternative division rings.
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| 10. | In contemporary mathematics, the division ring of quaternions exemplifies an algebra over a field.
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