‎we introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎we also introduce the concept of $(i,j)$-cohen--macaulay modules as a generalization of concept of cohen--macaulay modules‎. ‎these kind of modules are different from cohen--macaulay modules‎, as an example shows‎. ‎also an art...

We determine the simplicial complexes ∆ whose Stanley-Reisner ideals I∆ have the following property: for all n ≥ 1 the powers In ∆ have linear resolutions and finite length local cohomologies.

1. Primary ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Primary modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Primary decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. More ways to get associated primes . . . . . . . . . . . . . . . . . . . . . . . 12 5. Witnesses . . . . . . . . . . . . . . . . . . ...