Show that the center ZD of a division ring D is a fieldSolut

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.)