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2008-05-12T16:18:42Z

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Vipul: New page: ==Statement== ===Verbal statement=== Any Artinian ring is IZ: every element is either invertible, or a zero divisor. ==Proof== ''Given'': An Artinian ring $A$
2008-03-11T14:24:53Z

New page: ==Statement== ===Verbal statement=== Any Artinian ring is IZ: every element is either invertible, or a zero divisor. ==Proof== ''Given'': An Artinian ring <math>A</math...

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==Statement==

===Verbal statement===

Any [[Artinian ring]] is [[IZ-ring|IZ]]: every element is either invertible, or a zero divisor.

==Proof==

''Given'': An Artinian ring <math>A</math>, and an element <math>x \in A</math>

''To prove'': <math>x</math> is invertible or a zero divisor

''Proof'': Consider the descending chain of ideals:

<math>A \supset (x) \supset (x^2) \supset \ldots</math>

By the Artinianness, this chain stabilizes at some point, so we have:

<math>x^n = ax^{n+1}</math>

for some <math>a \in A</math>. Rewriting, we see that:

<math>x^n( 1 - ax) = 0</math>

If <math>1 - ax = 0</math>, then <math>x</math> is invertible. Otherwise, <math>x^n</math> is a zero divisor, and hence <math>x</math> is a zero divisor.