# Answer to Question #16671 in Linear Algebra for sanches

Question #16671

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.

Expert's answer

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*.
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