| 11. | Division rings differ from fields only in that their multiplication is not required to be commutative.
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| 12. | The Artin Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring.
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| 13. | However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields.
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| 14. | Historically, division rings were sometimes referred to as fields, while fields were called commutative fields .
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| 15. | Homogeneous coordinates for projective spaces can also be created with elements from a division ring ( skewfield ).
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| 16. | A commutative ring with unity satisfying the last condition is called a containment-division ring ( CDR ).
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| 17. | The Artin Zorn theorem generalizes the theorem to alternative rings : every finite alternative division ring is a field.
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| 18. | If " R " is a division ring or a field, then these are its only ideals.
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| 19. | The Artin Wedderburn theorem asserts that every semisimple ring is a finite product of full matrix rings over division rings.
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| 20. | The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring.
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