if S is infinite there is no good relation between property A(n) for R and property A(n) for the connected component rings of R (Examples 5.16 and 5.21). §5 consists of eight counterexamples, four of which have been mentioned above. Example 5.1 is a noncommutative ring which is not a left A(n) ring for any «. Three sufficient conditions for a ring R to be an A(l) ring are: (1) R is an A(0) ring; (2) R is a C-ring, meaning that any exact sequence 0->M-> F-> N ^0 of finitely generated 7?-modules with M flat and F free splits ; and (3) R is a 73-ring, meaning that for any index set 7, R' is a submodule of a flat module. Examples 5.2, 5.3, and 5.6 show that these three conditions are logically independent and hence that none of them are necessary for R to be an A(\) ring. Our notation is basically that of [3]; in addition we use "f.g." for finitely generated and "f.p." for finitely presented. ls is the identity function on a set S; » stands for all kinds of isomorphism; Si; is the Kronecker delta; Zis the ring of integers, and N is the set of positive integers. 2. General results. This section includes all those statements about A(n) rings which we know to be valid for rings which are not necessarily commutative. Propositions 2.1 and 2.2, as well as the essential ideas in the proofs of 2.3 and 2.4 are all due to M. Auslander. Proposition 2.3 and Theorem 2.4 were first proved by S. Endo [8] for the special case when R is commutative and T is a ring of quotients of R with respect to a multiplicative system of regular elements. 2.1. Proposition. Let R be a ring, M a left R-module. For each set I, put M' = \~\iei M, and let a(M): R1 g) M->■ M1 be the canonical homomorphism. Then (i) M is f.g. o a(M) is surjective,for all sets 7(l). (ii) M is f.p. o c(M) is bijective,for all sets I. Proof. First suppose o(M): RM ® M'-> MM is surjective. Then 3/j,..., fkeRM, and xx,.. .,xke M such (hat
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),
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