31 research outputs found
Finitely presented modules over semihereditary rings
One proves that each almost local-global semihereditary ring has the stacked
basis property and is almost Bezout. If M is a finitely presented module, its
torsion part tM is a direct sum of cyclic modules where the family of
annhilators is an ascending chain of invertible ideals. These ideals are
invariants of M. Moreover, M/tM is a direct sum of 2-generated ideals whose
product is an invariant of M. The idempotents and the positive integers defined
by the rank of M/tM are invariants of M too
Local rings of bounded module type are almost maximal valuation rings
It is shown that every commutative local ring of bounded module type is an
almost maximal valuation ring
Indecomposable injective modules of finite Malcev rank over local commutative rings
It is proven that each indecomposable injective module over a valuation
domain is polyserial if and only if each maximal immediate extension
of is of finite rank over the completion of
in the -topology. In this case, for each indecomposable injective module
, the following invariants are finite and equal: its Malcev rank, its
Fleischer rank and its dual Goldie dimension. Similar results are obtained for
chain rings satisfying some additional properties. It is also shown that each
indecomposable injective module over one Krull-dimensional local Noetherian
rings has finite Malcev rank. The preservation of Goldie dimension finiteness
by localization is investigated too
Weak dimension of FP-injective modules over chain rings
It is proven that the weak dimension of each FP-injective module over a chain
ring which is either Archimedean or not semicoherent is less or equal to 2
Valuation domains whose products of free modules are separable
It is proved that if is a valuation domain with maximal ideal and if
is countably generated for each prime ideal , then is separable
if and only is maximal, where
Commutative rings whose cotorsion modules are pure-injective
Let R be a ring (not necessarily commutative). A left R-module is said to be
cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that
each pure-injective left R-module is cotorsion, but the converse does not hold:
for instance, if R is left perfect but not left pure-semisimple then each left
R-module is cotorsion but there exist non-pure-injective left modules. The aim
of this paper is to describe the class C of commutative rings R for which each
cotorsion R-module is pure-injective. It is easy to see that C contains the
class of von Neumann regular rings and the one of pure-semisimple rings. We
prove that C is strictly contained in the class of locally pure-semisimple
rings. We state that a commutative ring R belongs to C if and only if R
verifies one of the following conditions: (1) R is coherent and each
pure-essential extension of R-modules is essential; (2) R is coherent and each
RD-essential extension of R-modules is essential; (3) any R-module M is
pure-injective if and only if Ext 1 R (R/A, M) = 0 for each pure ideal A of R
(Baer's criterion)
Modules with RD-composition series over a commutative ring
If R is a commutative ring, we prove that every finitely generated module has
a pure-composition series with indecomposable factors and any two such series
are isomorphic if and only if R is a Bezout ring and a CF-ring
Gaussian trivial ring extensions and fqp-rings
Let A be a commutative ring and E a non-zero A-module. Necessary and
sufficient conditions are given for the trivial ring extension R of A by E to
be either arithmetical or Gaussian. The possibility for R to be B{\'e}zout is
also studied, but a response is only given in the case where pSpec A (a
quotient space of Spec} A) is totally disconnected. Trivial ring extensions
which are fqp-rings are characterized only in the local case. To get a general
result we intoduce the class of fqf-rings satisfying a weaker property than
fqp-ring. Moreover, it is proven that the finitistic weak dimension of a
fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or infinite