Faithful Noetherian modules

Edward Formanek
1973 Proceedings of the American Mathematical Society  
The Eakin-Nagata theorem says that if T is a commutative Noetherian ring which is finitely generated as a module over a subring R, then R is also Noetherian. This paper proves a generalization of this result. However, the main interest is that the proof is very elementary and uses little more than the definition of "Noetherian". All rings are associative and have a unit, subrings have the same unit, and modules are unitary. A theorem due independently to Eakin [2] and Nagata [7] says that if
more » ... [7] says that if T=Rax+ • • -+Rak is a commutative ring finitely generated as a module over a subring R, then R is Noetherian if T is Noetherian. A later proof was given by Mollier [6] and there have been noncommutative generalizations by Eisenbud [3], Björk [1] and Jategaonkar and Formanek [4]. The object of this paper is to present a simple and elementary proof of the Eakin-Nagata theorem which generalizes the original version in a new direction. The proof is essentially a contraction of Eakin's proof as presented by Kaplansky in [5, p. 54], based on the observation that much ofthat proof disappears if one is not "handicapped" by the hypothesis that T is a ring. More precisely, T is viewed as an Rmodule and the Eakin-Nagata theorem is viewed as a generalization of the basic result that a commutative ring which has a faithful Noetherian module is itself Noetherian [5, Exercise 10, p. 53]. Theorem. Let R be a commutative ring and T=Rax + -• -+Rak a faithful finitely generated left R-module which satisfies the ascending chain condition on "extended submodules" AT, where A is an ideal in R. Then
doi:10.1090/s0002-9939-1973-0379477-x fatcat:ncq2gn5iejgwfjskq2wwv6z7y4