n-gr-coherent rings and Gorenstein graded modules

Let R be a graded ring and n ? 1 be an integer. We introduce and study the notions of Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules by using the notion of special finitely presented graded modules. On n-gr-coherent rings, we investigate the relationships between Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules. Among other results, we prove that any graded module in R-gr (or gr-R) admits a Gorenstein n-FP-gr-injective (or Gorenstein n-gr-flat) cover and preenvelope, respectively.

     

Anti-isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an anti-isomorphism of graded endomorphism rings of the strict gr-generators is induced by a graded Morita anti-equivalence or a graded anti-semilinear isomorphism.     

Isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an isomorphism of graded endomorphism rings of strong gr-generators is induced by a gr-generator, graded Morita equivalence, or semi-linear isomorphism. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 3–18, 2007.     

Cofiniteness and finiteness of local cohomology modules over regular local rings

Let (R,m) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAss _R (R/I) = Assh _R (_I). It is shown that the R- module H^ht(I) _I (R) is I-cofinite if and only if cd(I,R) = ht(I). Also we present a sufficient condition under which this condition the R-module H ⁱ _I (R) is finitely generated if and only if it vanishes.     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

Fragments with finiteness conditions in particular over group rings

Abstract length functions on groups were introduced by Lyndon [5]. In [2] Chiswell showed that for a group action on an IR-tree there is a length function associated with each point of the tree, and moreover any length function can occur in this way. This paper establishes necessary and sufficient conditions for two length functions to arise from the same action of a group on a tree.     

Isomorphisms and anti-isomorphisms of endomorphism rings of graded modules close to free ones

Doklady Mathematics -     

A commutativity and finiteness condition for rings

We prove that if a ring R has the property that XY = YX for all infinite subsets X and Y, then R is either finite or commutative.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.