Determinantal rank and flat modules

Abstract

By ring we mean commutative ring with identity. Module means unitary module. In this paper we use some results on determinantal rank to prove the following proposition: A finitely generated P-module M is projective if and only if M is flat and there is an exact sequence 0-*M-*N-*L of i?-modules such that N and L are projective (Theorem 2.9). A corollary is that a finitely generated R module M is proj ective if and only if M is flat, reflexive and M * = HomÄ (M, R) is of finite presentation. In §3, we give an example of a cyclic ideal M in a ring R such that M is flat and reflexive, M* is cyclic, but M is not projective. We use f.g. in place of finitely generated and morphism instead of P-homomorphism. The set of prime ideals of a ring R is denoted Spec(i?). N denotes the set of nonnegative integers. If SQN is unbounded, we write sup(S) = ». 2. Let u: M-^N be a morphism of P-modules. We define rk(z¿), the rank of u, by rk(«) =sup{nQN; \nu 9*0} where A" denotes «th exterior power. We also define dim(Af) =rk(ljvf). When M and N are f.g. free P-modules, rk(u) is also the determinantal rank of a matrix corresponding to u and dim (If) is the cardinality of a basis of M. When M and N are free we denote by D(u, p) the ideal generated by the ^-minors of a matrix corresponding to u. The ideals {D(u, p) ; p QN} are the Fitting invariants of Coker(w) [3]. If 5 is a multiplicative system in R, then rk(ws) is the rank of us as an Ps-morphism. The following result from [2, p. 98, Exercise 3] will be used several times. 2.1. Lemma. Let M and N be f.g. free R-modules of dimensions m and n respectively. Then a morphism u: Af-»A7 is a monomorphism if and only if m^n and Ann(D(u, m)) =0. (In that case rk(w) =m.) 2.2. Lemma. Let u: M->N be a morphism of R-modules. Let S and T be multiplicative systems in R such that S^T (i.e. R-*Rt factors through R-^Rs). Then (i) rk(ttr) ^rk (tts),

Rings for Which Certain Flat Modules are Projective

“…Left A(0) rings have also been studied by F. Sandomierski and D. E. Smith [20]. S. Cox has proved that every commutative ring is an A(2) ring [7,Theorem 2.9]. A(\) rings have been considered by M. Auslander, to whom the authors wish to express their gratitude for several helpful conversations.…”