Journal of Pure and Applied Algebra 39 (1986) 125-139 North-Holland 125 WHEN IS AN N-RING NOETHERIAN? William HEINZER Purdue University, West Lafayette, IN 47907, U.S.A. David LANTZ Colgate University, Hamilton, NY 13346, U.S.A. Communicated by C. Weibel Received 1 June 1984 1. Introduction In [4], Gilmer and Heinzer introduced the concepts of N-ring, N-domain, and N- domain for overrings: A ring R (always commutative with unity) is called an ‘N- ring’ if, for every ideal I of R, there is a Noetherian ring T containing R (with the same unity) from which Z is contracted, i.e., for which IT n R = I. If R and the T’s are required to be domains, then R is an ‘N-domain’; and if in addition the T’s are required to be contained in the quotient field of R, then R is an ‘N-domain for over- rings’. The paper [4] studied the question of whether an N-domain for overrings was always Noetherian. It proved the answer was “Yes” if the dimension of the N- domain for overrings was 1, but on the other hand it displayed N-rings which are not Noetherian. An example due to Hochster is described in [6]; it is a one-dimensional, quasilocal domain which is an N-ring but is not Noetherian. But despite a large number of positive results on these rings in [4] and a characterization of N-rings in [a], the question above still lacks a general answer. In Section 2 of the present paper we pro- vide variants of Hochster’s example to show that a domain which is an N-ring need not be an N-domain, and that a two-dimensional, quasilocal domain which is an N- ring need not be Noetherian. (We also see that another natural way to raise the dimension in Hochster’s example results in a loss of the N-ring property.) Several results of [4] show that an N-domain is a subring of Noetherian domains in natural ways. In Section 3 we provide some conditions under which the Noetherian property descends in such embeddings. For instance, a locally finite in- tersection of one-dimensional local domains is Noetherian if the residue field exten- sions are all finite. And an intersection R of a field and a one-dimensional local domain T is Noetherian if the residue field of T is finite over that of R; while the integral closure of R is Noetherian if the residue field extension is algebraic. The 0022-4049/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland)