In this article, singular d-homology, singular extended d-homology, relative singular d-homology and relative singular extended d-homology functors are introduced and some axioms, such as dimension axiom, homotopy axiom, excision axiom and exactness axiom, of homology theory relative to these homologies are investigated and are proved to hold.

In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present su cient conditions for the existence of long exact extended d-homology sequence. Also we give some illustrative examples.

We investigate the relative cohomology and relative homol-ogy theories of F -Gorenstein modules, consider the relations between classical and F -Gorenstein (co) homology theories.

We show some results about local homology modules and local cohomology modules concerning to being in a serre sub category of the category of R-modules. Also for an ideal I of R we define the concept of CI condition on a serre category, which seems dual to CI condition of Melkerson [1]. As a main result we show that for any minimax R-module M of any serre category S that satisifies CI (CI) condition the local homology module HiI(M) (local cohomology module HIi(M) 2 S) for all i ≥ 0.

Let R be a commutative Noetherian ring, a be an ideal of R, and D(R) denote the derived category of R-modules. For any homologically-bounded complex X, we conjecture that supL a(X)  magRX. We prove this in several cases.