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Rings for Which Certain Flat Modules are Projective

1970; American Mathematical Society; Volume: 150; Issue: 1 Linguagem: Inglês

10.2307/1995487

1088-6850

S. H. Cox, Rosemary L. Pendleton,

Advanced Topics in Algebra

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.Proof.Each Rt is a projective left /^module, so it follows easily that if R is an A(n) ring, n^O, so is each Z?{.Now suppose each Rt is a left A(n) ring.For each i e I, and each left .R-module N, let Ni = Ri Ex -> • • ■ -> En be an exact sequence of f.g.left Ä-modules with M flat and E} free, Vy.It is clear that Mt is /î-projective, V¿ e I. If/is finite, then M=@ M¡ is projective.Otherwise, let «^ 1, and consider the functor T which associates to each left A-module N the left .R-module \~I Nt.