Answer to Question #16671 in Linear Algebra for sanches

Question

Let K be a division ring with center k. For any a ∈ K\k, show that the ideal generated by x − a in K[x] is the unit ideal.

Answer 1

Fix b ∈ K such that ab (not equal) ba. Then (x − a) (the ideal generated by x − a) contains b(x − a) − (x − a)b = ab − ba ∈U(K), so (x − a) = R.