On the exactness of flat resolvents

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR < n ( n > 0) if and only if every flat resolvent 0 → M → F° → F¹... of a finitely presented right R—module M is exact at F'(i > n?1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) < n (n > 1) if and only if every (n?l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized     

Flat covers and cotorsion envelopes of sheaves

In this paper we prove that any sheaf of modules over any topological space (in fact, any -module where is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( -modules) does not have in general enough projectives.

     

Torsion-free covers and pure-injective envelopes over valuation domains

We describe the torsion-free covers of cyclic modules, the pure-injective envelopes of ideals, the maximal immediate extensions of localizations and the injective envelopes of cyclics over valuation domains. We study the relations among these modules. This paper generalizes some results of Banaschewski, Cheatham, Enochs and Nishi. Supported by Ministero della Pubblica Istruzione and GNSAGA (CNR).     

Gorenstein injective envelopes and covers over n-perfect rings

We prove the existence of Gorenstein injective envelopes and covers over n-perfect rings for some classes of modules associated with a dualizing bimodule.     

Torsionfree covers and covers by submodules of flat modules 1

Throughout this paper D denotes a division ring and V a left vector space over D. The finitary general linear group FGL(V) or FA Aut_DV over V is the subgroup of Aut_DV of D-automorphisms g of V such that [V,g] = V(g-l) has finite (left) dimension over D. By a finitary skew linear group we mean any subgroup G of FGL(V) for any D and V. Such a G is irreducible if V is irreducible as D-G (bi)module and is primitive if whenever V = ⊕_{ω ? Ω}V_omega as D-module, where for all g?G and ω?Ω, V_ωg = V_ω for some ω?Ω, we have |Ω| = 1. In [4] we showed that a primitive irreducible finitary skew linear group is finite dimensional if it is hyper locally nilpotent (that is radical in the sense of Kuros) and sometimes if it is locally soluble. Here we complete the locally soluble case and, in fact, we can be a little more general.     

Modules invariant under automorphisms of their covers and envelopes

In this paper we develop a general theory of modules which are invariant under automorphisms of their covers and envelopes. When applied to specific cases like injective envelopes, pure-injective envelopes, cotorsion envelopes, projective covers, or flat covers, these results extend and provide a much more succinct and clear proofs for various results existing in the literature. Our results are based on several key observations on the additive unit structure of von Neumann regular rings.     

Monomorphic flat envelopes in commutative rings

The authors have been partially supported by the C.A.I.C.Y.T.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

Divisibility classes are seldom closed under flat covers

We show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring $R/sR$ satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.     

The structure of commutative rings with monomorphic flat envelopes

It is obvious that OF and Von Neumann regular rings have monomorphic flat envelopes. In this paper we completely describe the structure,in terms of OF and Von Neumann regular rings, of those commutative rings all of whose modules have a monomorphic flat envelope (m.f.e. ). For that, we introduce the notion of locally QF ring with m.f.e., whose structure is given in terms of OF rings. It turns out that a commutative ring R with m.f.e. is characterized as a (essential) subdirect product of a locally QF ring with m.f.e. and a Von Neumann regular ring, with the latter flat as an R-module.     

Flows, flow-pair covers and cycle double covers

In this paper, some earlier results by Fleischner [H. Fleischner, Bipartizing matchings and Sabidussi’s compatibility conjecture, Discrete Math. 244 (2002) 77-82] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow parity-pair-cover of G if the union of their supports covers the entire graph; f₁ is an h-flow and f₂ is a k-flow, and . Then G admits a nowhere-zero 6-flow if and only if G admits a (4,3)-flow parity-pair-cover; and G admits a nowhere-zero 5-flow if G admits a (3,3)-flow parity-pair-cover. A pair of integer flows (D,f₁) and (D,f₂) is an (h,k)-flow even-disjoint-pair-cover of G if the union of their supports covers the entire graph, f₁ is an h-flow and f₂ is a k-flow, and for each {i,j}={1,2}. Then G has a 5-cycle double cover if G admits a (4,4)-flow even-disjoint-pair-cover; and G admits a (3,3)-flow parity-pair-cover if G has an orientable 5-cycle double cover.     

Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

We analyze the stability of Muckenhoupt's and classes of weights under a nonlinear operation, the -operation. We prove that the dyadic doubling reverse Hölder classes are not preserved under the -operation, but the dyadic doubling classes are preserved for . We give an application to the structure of resolvent sets of dyadic paraproduct operators.

     

Gorenstein Projective,Injective, and Flat Complexes

Abstract

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S?∈?Sy l ₂(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out _c (G)?=?1. If S is generalized quaternion, 𝒵(S)???𝒵(G) and S ₄ is not a homomorphic image of G, then Out _c (G)?=?1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.