In a division ring, every centralizer is a division ring

Prove that if D is a division ring, then C_D(a) is a division ring for all a \in D.

We saw in the previous exercise that C_D(a) is a subring which contains 1. It remains to be shown that every nonzero element of C_D(a) has an inverse in C_D(a). To that end, let x \in C_D(a) be nonzero. Then xa = ax. Now x^{-1} exists in D, and we have x^{-1}xa = x^{-1}ax, so that a = x^{-1}ax. Similarly, right multiplying by x^{-1} yields ax^{-1} = x^{-1}a. Thus x^{-1} \in C_D(a). Thus C_D(a) is a division ring.

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