Tate local cohomology and Gorenstein theory, which are important generalizations of the classical cohomology, has been investigated. It found that they have such vanishing properties long exact sequences. However, for homology, what about duality? In this paper we concerned with homology homology. first part generalize as study properties, artinianness some sequence modules. We find an artiania...

We study Gorenstein flat objects in the category Rep(Q,R) of representations a left rooted quiver Q with values Mod(R), all R-modules, where R is an arbitrary associative ring. show that representation X if and only for each vertex i canonical homomorphism φiX:⊕a:j→iX(j)→X(i) injective, R-modules X(i) CokerφiX are flat. As application, we obtain model structure on which give explicit descriptio...

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give sufficient condition for the regular locus to be open, show that is always open. However, both of these loci are often empty: no matter how nice H0(A) is, there examples where A empty. then DG-module contains dense open set....

Let (A,m) be a Noetherian local ring and F = (In)n≥0 a filtration. In this paper, we study the Gorenstein properties of the fiber cone F (F), where F is a Hilbert filtration. Suppose that F (F) and G(F) are CohenMacaulay. If in addition, the associated graded ring G(F) is Gorenstein; similarly to the I-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal nu...

Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. These also give examples of semiinjective complexes whose localization need not be homotopically injective. Let R be a commutative noetherian ring. Recall that the support of a finitely generated R-module M is the set of...

A Noetherian integral domain R is said to be a splinter if it is a direct summand, as an R–module, of every module–finite extension ring, see [Ma]. In the case that R contains the field of rational numbers, it is easily seen that R is splinter if and only if it is a normal ring, but the notion is more subtle for rings of characteristic p > 0. It is known that F–regular rings of characteristic p...

relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $m$, called {em $tau$-rank of} $m$, which coincides with the reduced rank of $m$ whenever $tau$ is the goldie torsion theory. it is shown that the $tau$-rank of $m$ is measured by the length of certain decompositions of the $tau$-injective hull of $m$. moreover, some relations between the $tau$-rank of $m$ and c...

We give a criterion for the existence of a super-decomposable pure-injective module over an arbitrary serial ring.

It is shown that a ring R is (von Neumann) regular if and only if every cyclic left R-module is GP -injective if and only if R is left PP and left GP -injective. In addition, two examples are given to answer two open questions in the negative. 2000 Mathematics Subject Classification: 16E50, 16D60