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### Let $D$ be a UFD such that Bezout's Identity holds. Then Every ideal is finitely generated implies that $D$ is PID [duplicate]

My proof $I=<a_1,\dots,a_n>=Da_1+Da_2+\cdots +Da_n$ let $g=gcd(a_1,\dots,a_n)$ Then since Bezout's identity holds and the binary operator gcd is associative, $g\in I$. Also any element of $I$ is ...
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### Ring whose finitely generated ideals are principal [duplicate]

The boolean ring $\prod_{n\in N} (Z/2Z)$ is an example of rings that verifie two properties every finitely generated ideals are principal there exits ideals that are not principal. My question : is ...
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### Constructing nonprincipal ideals in a non-UFD

It's well-known that all PIDs are UFDs, i.e. all non-UFDs are not PIDs. Now, it seems to me that there are two ways that a ring $R$ could fail to be a UFD: Some element $x$ has no factorisation into ...
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Let $D$ be an integral domain such that for any $a,b \in D$, $Da+Db$ is a principal ideal. Then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ?