For which rings can elements be written as a finite product of irreducible elements?

Asked Oct 06 '16 at 19:07

Active Oct 06 '16 at 21:20

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For which rings $A$ can elements be written as a finite product of irreducible elements?

I feel like this is true for almost all rings I have ever encountered, but when do I know that I can write any nonzero nonunit element of $A$ as a finite product of irreducible elements? (I am not assuming the product is unique here.)

In particular, if $A$ is an affine $k$-algebra can I always do this? ($k$ algebraically closed field)

Thanks!

edited Oct 06 '16 at 21:20

user26857

52,094

asked Oct 06 '16 at 19:07

Johnny T.

2,897

1

For a Noetherian ring the answer is yes. – Matt Samuel Oct 06 '16 at 19:11
How would one go about proving it? Does it follow easily from the definition? – Johnny T. Oct 06 '16 at 19:16
1

@Johnny For a rough sketch, factoring out elements successively gives you an ascending chain of principal ideals. When the chain stabilizes the element is irreducible. – Matt Samuel Oct 06 '16 at 19:18

For which rings can elements be written as a finite product of irreducible elements?

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