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Vipul https://commalg.subwiki.org/w/index.php?title=Noetherian_and_Bezout_iff_principal_ideal&diff=809&oldid=prev 2008-01-07T23:34:06Z

<p></p> <p><b>New page</b></p><div>==Statement==<br /> <br /> A ring is a [[principal ideal ring]] if and only if it is both a [[Noetherian ring]] and a [[Bezout ring]].<br /> <br /> In particular, an [[integral domain]] is a [[principal ideal domain]] if and only if it is both a [[Noetherian domain]] and a [[Bezout domain]].<br /> <br /> ==Definitions used==<br /> <br /> A ring is a:<br /> <br /> * [[principal ideal ring]] if &#039;&#039;every&#039;&#039; [[ideal]] is a [[principal ideal]]<br /> * [[Noetherian ring]] if &#039;&#039;every&#039;&#039; [[ideal]] is a [[finitely generated ideal]]<br /> * [[Bezout ring]] if every [[finitely generated ideal]] is a [[principal ideal]]<br /> <br /> ==Proof==<br /> <br /> The fact that a Noetherian Bezout ring is a principal ideal ring follows by a single line of logic based on the above definitions. The fact that every principal ideal ring is Bezout is again direct from the definitions. The fact that it is Noetherian follows from the fact that a principal ideal is, by definition, finitely generated.</div>

Vipul