I am at a very initial stage of commutative algebra.

Yesenia Obrien 2022-07-07 Answered
I am at a very initial stage of commutative algebra. I want to know whether the power of a prime ideal in a commutative ring is prime ideal or not?
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Answers (1)

gutinyalk
Answered 2022-07-08 Author has 11 answers
An ideal I R is prime iff R / I is an integral domain (that is, it contains no nontrivial zero divisors).
If I is a prime ideal such that I 2 I, then in the quotient ring R / I 2 the elements of the form x + I 2 , where x I, will be nilpotents. Thus, R / I 2 is not a domain and I 2 is not prime.
For a concrete and illustrative example, consider I = ( p ) Z - the ideal of the ring of integers generated by a prime p. Then, I 2 = ( p 2 ), and is not prime (since it is generated by a non-prime integer).
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