mathmari
Gold Member
MHB
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Hey! 
Let $R$ be an integral domain and a Noetherian U.F.D. with the following property:
for each couple $a,b\in R$ that are not both $0$, and that have no common prime divisor, there are elements $u,v\in R$ such that $au+bv=1$.
I want to show that $R$ is a P.I.D..
Since $R$ is Noetherian, we have that every ideal is finitely generated. Then every ideal is a finite product of irreducible elements of $R$.
How could we use the above property of $R$ to conclude that $R$ is a P.I.D. ? (Wondering)

Let $R$ be an integral domain and a Noetherian U.F.D. with the following property:
for each couple $a,b\in R$ that are not both $0$, and that have no common prime divisor, there are elements $u,v\in R$ such that $au+bv=1$.
I want to show that $R$ is a P.I.D..
Since $R$ is Noetherian, we have that every ideal is finitely generated. Then every ideal is a finite product of irreducible elements of $R$.
How could we use the above property of $R$ to conclude that $R$ is a P.I.D. ? (Wondering)