Show that the center Z(D) of a division ring D is a field. Solution By defintion, a division algebra D over a field F is a vector space over F with a product operation, which satisfies all the usual laws of associativity and distributivity (but not necessaily commutative). in addition to the property that evey non-zero element in D is invertible. Now Z(D) is the center of the division algebra = {x in D~ xy =yx}, the set of all x in D commuting with all the elements of D. Clearly Z(D) itself is a division algebra . ( xy =yx implies y-1 x-1 = x-1 y-1 for all y implies Z(D) is closed under inverses). By very definition Z(D) is a commutative . So Z(D) is a field. (any commutaive division algebra is a field.).

1. Show that the center Z(D) of a division ring D is a field.
Solution
By defintion, a division algebra D over a field F is a vector space over F with a product
operation,
which satisfies all the usual laws of associativity and distributivity (but not necessaily
commutative).
in addition to the property that evey non-zero element in D is invertible.
Now Z(D) is the center of the division algebra = {x in D~ xy =yx}, the set of all x in D
commuting with
all the elements of D.
Clearly Z(D) itself is a division algebra . ( xy =yx implies y-1 x-1 = x-1 y-1 for all y implies
Z(D) is closed
under inverses).
By very definition Z(D) is a commutative .
So Z(D) is a field. (any commutaive division algebra is a field.)