Abstrakt
Let C be a class of modules and L = limRIGHTWARDS ARROW C the class of all direct limits of modules from C. The class L is well understood when C consists of finitely presented modules: L then enjoys various closure properties. Our first goal here is to study the closure properties of L in the general case when C SUBSET OF OR EQUAL TO Mod-R is arbitrary. Then we concentrate on two important particular cases, when C = add M and C = Add M, for an arbitrary module M.
In the first case, we prove that lim
RIGHTWARDS ARROW add M = {N ELEMENT OF Mod-R |
THERE EXISTS F ELEMENT OF FS : N TILDE OPERATOR+D91= F CIRCLED TIMES S M} where S = End M, and
FS is the class of all flat right S-modules. In the second case, lim
RIGHTWARDS ARROW Add M = {F S M | F ELEMENT OF FS} where S is the endomorphism ring of M endowed with the finite topology,
FS is the class of all right S-contramodules that are direct limits of direct systems of projective right S-contramodules, and F S M is the contratensor product of the right Scontramodule F with the discrete left S-module M.
For various classes of modules D, we show that if M ELEMENT OF D then lim
RIGHTWARDS ARROW add M = limRIGHTWARDS ARROW Add M (e.g., when D consists of pure projective modules), but the equality for an arbitrary module
M remains open. Finally, we deal with the question of whether lim
RIGHTWARDS ARROW Add M = AddM where AddM is the class of all pure epimorphic images of direct sums of copies of a module M.