Principal ideal domain

Maximal ideal iff prime ideal in a PID

Let be a PID, and be a nonzero ideal. Then is prime iff it is maximal.¹

Proof

A maximal ideal in a commutative ring is prime. For the converse, suppose is a prime ideal with , and suppose for some . It follows for some , so from primality of we have or . If it follows . If it follows for some , so

so from cancellation , so .

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Footnotes

1. 2009. Algebra: Chapter 0, §III.4.3, ¶4.13, pp. 151–152 ↩