[Math] Show that the center of a division ring is a field.

abstract-algebraring-theory

I tried the proof, but wasn't able to proceed further:
let a and b be any elements belonging to Z(center),
Now a.b=b.a
hence Z is commutative.

Now proving that Z is a division ring:(then we can show that since every commutative division ring is field, Z is a field)
NO CLUE…

Best Answer

Hopefully you can already prove:

The center of a ring is a ring, in fact, a commutative ring.

Furthermore, if $D$ is a division ring, then for all $x\in Z(D)$, if $x\neq 0$, then $x^{-1}$ exists somewhere in $D$.

Now to show the commutative ring $Z(D)$ is a field, you'd have to show that $x^{-1}\in Z(D)$, because inverses are unique, and a field necessarily has inverses for its nonzero elements.

So, the task is clear: if $0\neq x\in Z(D)$, prove $x^{-1}\in Z(D)$.

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