21. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring . 22. Lastly, there is even an example of a domain in a division ring which satisfies " neither" 23. According to Wedderburn's little theorem, any finite division ring must be commutative, and hence a finite field. 24. From the bundle theorem follows the existence of a ) a skewfield ( division ring ) and b ) an ovoid. 25. The Artin Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. 26. The Artin Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. 27. The Artin Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring . 28. Beginning with division rings arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. 29. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers.30. Failure of one of the two distributive laws brings about near-rings and near-fields instead of rings and division rings respectively.
