7,024 research outputs found
On certain homological finiteness conditions
In this paper, we show that the injective dimension of all projective modules over a countable ring is bounded by the self-injective dimension of the ring. We also examine the extent to which the flat length of all injective modules is bounded by the flat length of an injective cogenerator. To that end, we study the relation between these finiteness conditions
on the ring and certain properties of the (strict) Mittag-Leffler modules. We also examine the relation between the self-injective dimension of the integral group ring of a group and Ikenaga’s generalized (co-)homological dimensio
RD-flatness and RD-injectivity
It is proved that every commutative ring whose RD-injective modules are
-RD-injective is the product of a pure semi-simple ring and a finite
ring. A complete characterization of commutative rings for which each artinian
(respectively simple) module is RD-injective, is given. These results can be
obtained by using the properties of RD-flat modules and RD-coflat modules which
are respectively the RD-relativization of flat modules and fp-injective
modules. It is also shown that a commutative ring is perfect if and only if
each RD-flat module is RD-projective.Comment: A new section is added to the version published in Communications in
Algebra where a complete proof of Theorem 3.1 is give
Relative FP-injective and FP-flat complexes and their model structures
In this paper, we introduce the notions of -injective and -flat complexes in terms of complexes of type . We show that
some characterizations analogous to that of injective, FP-injective and flat
complexes exist for -injective and -flat complexes. We
also introduce and study -injective and -flat
dimensions of modules and complexes, and give a relation between them in terms
of Pontrjagin duality. The existence of pre-envelopes and covers in this
setting is discussed, and we prove that any complex has an -flat
cover and an -flat pre-envelope, and in the case that
any complex has an -injective cover and an -injective
pre-envelope. Finally, we construct model structures on the category of
complexes from the classes of modules with bounded -injective and
-flat dimensions, and analyze several conditions under which it is
possible to connect these model structures via Quillen functors and Quillen
equivalences.Comment: 41 page
The Existence of Relative pure Injective Envelopes
Let be a class of finitely presented -modules such that
and has a subset with the
property that for any there is a with
We show that the class of -pure injective
-modules is preenveloping. As an application, we deduce that the left global
-pure projective dimension of is equal to its left global
-pure injective dimension. As our main result, we prove that, in
fact, the class of -pure injective -modules is enveloping.Comment: to appear in Colloquium Mathematicu
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