Flat R-module Research Articles

The existence of flat covers over Noetherian rings of finite Krull dimension

Bass characterized the rings R with the property that every left R-module has a projective cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective R-modules coincides with the class of flat R-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring R, every left R-module has a flat cover. Known classes of rings for which the conjecture holds include Von Neumann regular rings (trivially), the left perfect rings (Bass), Prufer domains (Enochs), and then more generally, all right coherent rings of finite weak global dimension (Belshoff, Enochs, Xu). In this paper we show that the conjecture holds for all commutative Noetherian rings of finite Krull dimension and so for all local rings and all coordinate rings of affine algebraic varieties.

Rings with flat socles

We show that the class of rings with flat socles is closed under the formation of polynomial extensions, direct products, and excellent extensions. We also point out that if a ring R can be embedded into a semisimple Artinian ring, then all finitely generated flat R-modules are projective.

Strongly graded left FTF rings

An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if Re is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.

Covers by flat modules and submodules of flat modules

If R is a right coherent ring, then left R-modules have covers by submodules of flat R-modules if and only if all injective left R-modules have flat covers. This is the case if R is commutative and noetherian. If, furthermore, the Krull dimension of R is finite, then all cotorsion R-modules have flat covers. In this case, if F is a flat R-module, then F[[ x]] is a flat R[[ x]]-module.

Exoteric homomorphisms

R is a commutative reduced ring. If X⊆R,rR(X)={r∈R:Xr=0}. A ring homomorphism f : R → S is an exoteric homomorphism if for all pairs (I, J) of finitely generated ideals of R, rR(I) = rR(J) implies rS(f(I)) = rS(f(J)). Rings for which Qmax(R), the maximal quotient ring of R, is a flat R-module and rings which contain no finitely generated dense ideals are amongst the classes of rings which may be characterised through these homomorphisms.

Commutative semi-coherent and semi-regular rings

Throughout this paper all rings considered will be commutative rings with identity. We shall consider the consequences for a ring of the flatness of certain classes of its modules. Since flatness is determined by the finitely generated ideals of a ring we shall in a real sense be studying conditions on these ideals. A coherent ring is a ring whose finitely generated ideals are finitely presented. Chase has proved [3, Theorem 2.11 that a ring is coherent if and only if any direct product of its flat modules is flat. In [8, Theorem 1 J we proved that a ring R is coherent if and only if Hom,(B, C) is flat whenever B and C are injective R-modules. We define a ring R to be semicoherent if Hom,(B, C) is a submodule of a flat R-module whenever B and C are injective R-modules. This is a property shared by both integral domains and Noetherian rings. We prove in Proposition 1.1 that a reduced ring R is semi-coherent if and only if min R is compact. Proposition 1.3 states that if R is a semi-coherent ring, then a finitely generated submodule of a free R-module is flat if and only if it is projective. A regular ring is a ring whose finitely generated ideals are direct summands of the ring. Equivalently it is a ring such that every one of its modules is flat. Inspired by our definition of semi-coherent we define a ring to be semi-regular if every module is a submodule of a flat module. Proposition 2.2 states that if the total ring of quotients of a ring R is semiregular, then R is a semi-coherent ring. An even stronger connection with the notion of coherence is provided by Proposition 2.3: A ring R is semiregular if and only if R is coherent and RM is semi-regular for every maximal ideal M of R. Proposition 2.7 states that if R is a reduced ring, then R is semi-regular if and only if R is regular; and Proposition 3.4 states that if R is a Noetherian

Commutative Coherent Rings

Throughout this paper R will be a commutative ring with 1. The purpose of this paper is to provide two new characterizations of coherent rings. The first of these characterizations shows that the class of coherent rings is precisely the class of rings for which certain duality homomorphisms are isomorphisms. And the second of these characterizations shows that the class of coherent rings is precisely the class of rings for which the endomorphism ring of any infective module is a flat module. We can show as a consequence that the endomorphism ring of a universal infective R-module is a faithfully flat R-module whenever R is a coherent ring.

The Finiteness of I When R[ X ]/I is Flat

Let R be a commutative ring with identity, let X be an indeterminate, and let I be an ideal of the polynomial ring R[X]. Let min I denote the set of elements of I of minimal degree and assume henceforth that min I contains a regular element. Then R[XI/I is a flat R-module implies I is a finitely generated ideal. Under the additional hypothesis that R is quasi-local integrally closed, the stronger conclusion that I is principal holds. (An example shows that the first statement is no longer valid when min I does not contain a regular element.) Let c(I) denote the content ideal of I, i.e. c(I) is the ideal of R generated by the coefficients of the elements of I. A corollary to the above theorem asserts that R[X]/I is a flat R-module if and only if I is an invertible ideal of R[X] and c(I) = R. Moreover, if R is quasi-local integrally closed, then the following are equivalent: (i) R[X1/I is a flat R-module; (ii) R[X]/I is a torsion free R-module and c(I) = R; (iii) I is principal and c(I) = R. Let 6 denote the equivalence class of X in R[XI/I, and let (1, 6 . 4t denote the R-module generated by 1, 6, ***, ft. The following statements are also equivalent: (i) (1, 4, .., t) is flat for all t ? 0; (ii) (1, , ..., t) is flat for some t > 0 for which 1 -, .. t are linearly dependent over R; (iii) I = (f , , fn), fi e min I, and c(I) R; (iv) c(min I) = R. Moreover, if R is integrally closed, these are equivalent to R[X]/I being a flat R-module. A certain symmetry enters in when f is regular in R[L], and in this case (i)-(iv) are also equivalent to the assertion that R[L] and R[1/f are flat R-modules. The main results of this paper are found in ??2 and 3. Many of the difficulties of these sections already occur when the ring R is an integral domain, and the reader might benefit by first confining his attention to this case. Additional technical difficulties arise when one proceeds to the case that R is an arbitrary ring and I is subject to the restriction that min I contain a regular element. ?4 is devoted to a discussion of what happens when one removes this condition on min I. In particular, we make there a conjecture as to the class of rings R with the property that for every ideal I of R[X], if R[X]/I is R-flat, then I is finitely generated. The corresponding question for finitely generated modules is easily answered as follows: The class of rings R with the property that R[X]/I is a finite flat R-module implies I is finitely generated is exactly Presented to the Society, March 27, 1971; received by the editors May 10,1971. AMS 1970 subject classifications. Primary 13A15, 13B25, 13C10; Secondary 13B20.

Some homological results on certain finite ring extensions

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.

The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-flat. II

This paper supplements work of Ohm-Rush. A question which was raised by them is whether R [ X ] / I R[X]/I is a flat R-module implies I is locally finitely generated at primes of R [ X ] R[X] . Here R is a commutative ring with identity, X is an indeterminate, and I is an ideal of R [ X ] R[X] . It is shown that this is indeed the case, and it then follows easily that I is even locally principal at primes of R [ X ] R[X] . Ohm-Rush have also observed that a ring R with the property “ R [ X ] / I R[X]/I is R-flat implies I is finitely generated” is necessarily an A ( 0 ) A(0) ring, i.e. a ring such that finitely generated flat modules are projective; and they have asked whether conversely any A ( 0 ) A(0) ring has this property. An example is given to show that this conjecture needs some tightening. Finally, a theorem of Ohm-Rush is applied to prove that any R with only finitely many minimal primes has the property that R [ X ] / I R[X]/I is R-flat implies I is finitely generated.