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unique factorization domain

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Let R be a unique factorization domain then show that prime element R generates a prime ideal.

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Solution Summary

The solution analyzes prime elements for R generates. Factorization is examined.

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Proof:
Let us consider an arbitrary prime element r in R and the ideal I = <r> generated by r. We want to show that I is an prime ideal. We think about two arbitrary elements a and b in R with ab in I.
Since R is an unique factorization domain (UFD), then we have unique factorizations for both a and b
a = p_1 * p_2 * ...

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