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Vipul https://commalg.subwiki.org/w/index.php?title=Artinian_implies_Cohen-Macaulay&diff=47&oldid=prev 2008-03-11T14:19:16Z

<p>New page: {{curing property implication}} ==Statement== ===Property-theoretic statement=== The <a href="/wiki/Property_of_commutative_unital_rings" title="Property of commutative unital rings">property of commutative unital rings</a> of being <a href="/wiki/Artinian_ring" title="Artinian ring">Artinian</a> is stronger than the pro...</p> <p><b>New page</b></p><div>{{curing property implication}}<br /> <br /> ==Statement==<br /> <br /> ===Property-theoretic statement===<br /> The [[property of commutative unital rings]] of being [[Artinian ring|Artinian]] is stronger than the property of being [[Cohen-Macaulay ring|Cohen-Macaulay]].<br /> <br /> ===Verbal statement===<br /> <br /> Any [[Artinian ring]] is [[Cohen-Macaulay ring|Cohen-Macaulay]].<br /> <br /> ==Definitions used==<br /> <br /> ===Artinian ring===<br /> {{further|[[Artinian ring]]}}<br /> <br /> An Artinian ring is a commutative unital ring in which any descending chain of ideals stabilizes after a finite stage.<br /> <br /> ===Cohen-Macaulay ring===<br /> {{further|[[Cohen-Macaulay ring]]}}<br /> <br /> A Cohen-Macaulay ring is a ring in which, for every maximal ideal, the [[depth]] equals the [[codimension]].<br /> <br /> ==Facts used==<br /> <br /> * [[Artinian implies IZ|In an Artinian ring every element is invertible or a zero divisor]]: This follows by constructing the descending chain of principal ideals generated by powers of the element.<br /> <br /> ==Proof==<br /> <br /> In an Artinian ring, every prime ideal is maximal, so in particular Artinian rings are [[zero-dimensional ring|zero-dimensional]]. Thus, the codimension of any maximal ideal is zero. Hence, we need to prove that the depth of any maximal ideal is zero.<br /> <br /> The trick here is to use the Artinianness condition to show that &#039;&#039;every&#039;&#039; element of the ring is either invertible or a zero divisor. In particular, any element contained in a maximal ideal must be a zero divisor, and hence, there cannot be any regular sequence of positive length in a maximal ideal.</div>

Vipul