… For an R-module M, E(M) denotes the injective envelope of M, the character module Hom ℤ
… global dimension of a ring R. fd(M) and id(M) denote the flat and injective dimensions of M, …

… we introduce flat complexes. We prove that over a commutative noetherian ring with finite
Krull dimension any complex has a flat cover. We get similar results with DG-flat covers. These …

… commutative Noetherian ring of finite Krull dimension has a flat cover. This makes it possible
to apply … Then we revisit the existing envelopes and covers such as injective envelopes and …

… flat cover of the cotorsion envelope of M is isomorphic to the cotorsion envelope of the flat
cover … Let 𝒫 n (ℱ n ) be the class of all modules with projective (flat) dimension less than or …

… flat modules in the category of left R-modules, an F-cover is called a flat cover--or just a
cover … and of finite Krull dimension, then all R-modules have flat covers. There are several …

… covers and of cotorsion envelopes in the sense of Enochs, Warfield and Matlis. We also
construct approximations by modules of finite homological dimensions … the Flat Cover Conjecture…

… flat envelope we know that R is right coherent by [1, Proposition 2] and M* has a flat envelope,
and so M** has a projective cover … , and hence M has a projective cover. In fact, let F —» M …

… prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module
is at most the flat dimension of the injective envelope of R R. Then we get that the injective …

… exist over commutative noetherian rings of finite Krull dimension. In [13] Xu gave a well-…
of injective envelopes and the existence of flat covers adds to the many instances where ‘flat’ …

… covers and envelopes are primarily investigated for cotorsion pairs, in dealing with projective
dimension 1 covers and divisible envelopes … modules admit strongly flat covers — this was …