Prime Ideals For Commutative Rings
An ideal P of a commutative ring R is prime if it has the following two properties:
- If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P,
- P is not equal to the whole ring R.
This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say
- A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.
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—Unknown. Young Beichan (l. 61–64)