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Math Help - Primes and Irreducible in UFD

  1. #1
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    Primes and Irreducible in UFD

    Why is every irreducible element in UFD a prime element?

    x is irreducible means if x =ab, then either a or b is unit.

    x is prime means if x =ab, x|a or x|b.

    Now, let x be irreducible, let x=ab, WLOG let a be a unit.

    Since x is in UFD, let  x= u_{1} \cdot \cdot \cdot u_{r} where each u is irreducible.
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    If x is an irreducible and x divides a*b. If x|a then proof is complete, otherwise the only common factors are units and so (a,x) = R (ideal generated by a and x). Thus, (ab,bx) = (b). Now argue that (b) is a subset of (x) and so x divides b.
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