A plane defined over a non-commutative division ring ( a division ring that is not a field ) would therefore be Desarguesian but not Pappian.
42.
In general, Schur's lemma cannot be reversed : there exist modules that are not simple, yet their endomorphism algebra is a division ring.
43.
*PM : hermitian form over a division ring, id = 7626-- WP guess : hermitian form over a division ring-- Status:
44.
*PM : hermitian form over a division ring, id = 7626-- WP guess : hermitian form over a division ring-- Status:
45.
This can be restated as saying that the subspaces in the projective geometry correspond to the principal right ideals of a matrix algebra over a division ring.
46.
The projective planes for any field ( or, more generally, for every division ring ( skewfield ) isomorphic to its dual ) are self-dual.
47.
As a finite division ring is a finite field by Wedderburn's theorem, the order of the kernel of a finite quasifield is a prime power.
48.
A division ring K is equation x ^ 2 + ax + b = 0, \ a, b \ in K has at most two solutions.
49.
However, for dimension two, there are examples that satisfy these axioms that can not be constructed from vector spaces ( or even modules over division rings ).
50.
However, due to Wedderburn's little theorem, which states that all " finite " division rings are fields, all " finite"
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