When is 𝐹[𝑥,𝑦] a unique factorization domain?

Abstract

Although the commutative polynomial ring F [ x , y ] F[x,y] is a unique factorization domain (UFD) and the free associative algebra F ⟨ x , y ⟩ F\langle x,y\rangle is a similarity-UFD when F F is a (commutative) field, it is shown that the polynomial ring F [ x , y ] F[x,y] in two commuting indeterminates is not a UFD in any reasonable sense when F F is the skew field of rational quaternions.