A new characterization of GCD domains of formal power series

Abstract

By using the v v -operation, a new characterization of the property for a power series ring to be a GCD domain is discussed. It is shown that if D D is a UFD \operatorname {UFD} , then D\lBrack X\rBrack is a GCD domain if and only if for any two integral v v -invertible v v -ideals I I and J J of D\lBrack X\rBrack such that ( I J ) 0 ≠ ( 0 ) , (IJ)_{0}\neq (0), we have ( ( I J ) 0 ) v ((IJ)_{0})_{v} = ( ( I J ) v ) 0 , = ((IJ)_{v})_{0}, where I 0 = { f ( 0 ) ∣ f ∈ I } I_0=\{f(0) \mid f\in I\} . This shows that if D D is a GCD domain such that D\lBrack X\rBrack is a π \pi -domain, then D\lBrack X\rBrack is a GCD domain.