CHAPTER IX - Further Theory of Ideals in Commutative Rings

Summary

Primary ideals. Throughout this chapter we shall consider commutative rings only.

It was pointed out in Chapter V that in some important respects the prime ideals in an arbitrary commutative ring play a role similar to that of the primes in the ring I of integers. We now study a more general class of ideals which will be seen to bear roughly the same relation to the powers of a prime integer that the prime ideals do to the primes themselves.

An ideal q in the commutative ring R is said to be primary if ab ≡ 0(q), a ≢ 0(q), imply that bⁱ ≡ 0(q) for some positive integer i. As a simple illustration, it is clear that in the ring I every ideal (pⁿ), where p is a prime and n a positive integer, is primary. Furthermore, R is always a primary ideal, and (0) is primary if and only if every divisor of zero in R is nilpotent. Any prime ideal is also primary, and thus the concept of primary ideal may be considered as a natural generalization of that of prime ideal.