Some homological results on certain finite ring extensions

Abstract

All rings are commutative with identity and all modules are unitary. A ring R is connected if 0 and 1 are the only idempotent elements of R. R is semiconnected if the number of idempotents in R is finite. PROPOSITION. Suppose that R is connected, that I is a principal ideal of R[x], and that R[x]/I is a finitely generated R-module. Then R[x]/I is a free R-module. PROPOSITION. Suppose that R is semiconnected, that I is a principal ideal of R [x], and that R [x]'/I is a finitely generated Rmodule. Then R [x]/I is a projective R-module. These results are applied to integral extensions. Introduction. Let R be a commutative ring with identity, let x be an indeterminate, let I be an ideal of R[x], and let c(I) denote the ideal of R generated by the coefficients of the elements of I. The purpose of this article is to establish Proposition 4 (6) which states that under suitable conditions R [x]/I is a free (projective) R-module. These results are related to a previous result of Nagata which ensures the flatness of R[x]/I. The motivation for our work is the case where R is a field and where I is the ideal of R[x] which is generated by the minimal polynomial for an element which is algebraic over R. All rings are commutative with identity, and act on modules from the right. Terminology and notation in this paper are consistent with [2]. This research was supported by a grant from the National Research Council of Canada. LEMMA 1. Suppose that I is a principal ideal of R[x] and that c(I) =R; then R[x]/I is a flat R-module. PROOF. [3, Theorem 1]. DEFINITION 2. Let R be a ring. R is called connected if 0 and 1 are the only elements of R which are idempotent. A ring is connected if and only if its spectrum is connected as a topological space. Received by the editors January 25, 1972. AMS (MOS) subject classifications (1970). Primary 16A50.