Irreducible elements

  • Thread starter mansi
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  • #1
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Here’s an interesting question…
Let R be a commutative ring and ‘a’ an element in R. If the principal ideal Ra is a maximal ideal of R then show that ‘a’ is an irreducible element.
If a is prime, this is pretty obvious…if a is not prime, then we say a= bc for some b,c in R. Now we need to show that either of them is a unit. I can’t imagine how… :frown:
 

Answers and Replies

  • #2
Hurkyl
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If a is reducible, try finding a bigger ideal than (a). Maximal ideals are always prime ideals!
 
  • #3
mathwonk
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the usual definition of a prime element in a commutative ring with identity, is that it generates a prime ideal.
 

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