MINIMAL PRIMES OF IDEALS AND INTEGRAL RING EXTENSIONS

It is shown that if R is a commutative ring with identity having the property that ideals in R have only a finite number of minimal primes, then a finite A-algebra again has this property. It is also shown that an almost finite integral extension of a noetherian integral domain has noetherian prime spectrum. If a is an ideal in a ring R (tacitly assumed to be commutative with identity) and F is a prime ideal of R containing a, then F is called a minimal prime of a if there is no prime ideal of R containing a and properly contained in P. The ring R is said to have FC (for finite components) if each ideal of R has only a finite number of minimal primes. I am indebted to Professor Nagata for suggestions which helped me in obtaining the proofs of the theorems in this article, and I would like to thank him for his generous help. The special case in Theorem 1 where R is an integrally closed domain