On the Noetherian-like rings of E. G. Evans

Abstract

It is shown that if a commutative ring with identity R is nonnoetherian, then the polynomial ring in one indeterminate over R has an with infinitely many maximal prime divisors (in the sense of Nagata). Let R denote a commutative ring with 1, and for any A of R, let f(A)={rcE R|there exists s E R\A such that rs E A}. (By ideal we shall always mean ideal$R. The notation R\A denotes the set-complement of A in R.) .f(A) is merely the set of zero-divisors on the R-module RIA and is always a union of prime ideals of R. Evans [1] calls R a ZDring (zero-divisor ring) if for any A of R, .f(A) is a union of finitely many prime ideals. We shall prove here the following: THEOREM. R is noetherian if (and only if) the polynomial ring in one indeterminate R [X] is a ZD-ring. Evans has proved in [1] the 2 indeterminate analogue of this theorem (which follows from the theorem) and the special case of the theorem for R containing an infinite field. A prime P of R such that P is maximal with respect to the property of being contained in .f(A) is called a maximal N-prime (for Nagataprime) of A. Note that such a prime contains A and that .f(A) is the union of the maximal N-primes of A. (See [2] and [4] for a perspective on the associated primes of an ideal.) PROOF OF THEOREM. Suppose R is not noetherian. Then there exists a strictly ascending chain (0) < (a1) < (a1, a2) <* . < (a1, * * * , an) <. * of ideals of R. Let fo = X, fi = 1 + X, * . , fi = 1+f0fj * fi-l,,** . We wish to show that the A=(alfl, a2f1f2,.. * , anfi * *fn, * * )in R[X] has an infinite number of maximal N-primes and hence has the property that &t(A) is not a finite union of prime ideals. We show first that eachfi E (A). Since A c (fr) and f, is a monic polynomial of positive degree in R [X], it follows that A nfR= (0). Hence a1 0 A, so a1f, E A implies thatf1 E &t(A). Similarly, to show fn E 9(A), we wish to show anfi * fn-l 0 A. Consider Received by the editors September 16, 1971. AMS 1970 subject classifications. Primary 13E05, 13F20.