Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. It is unknown whether or not odd perfect numbers exist, although many conditions necessary for their existence have been found. For example, Cohen and Hagis ha...

If V is an equidimensional codimension c subscheme of an n-dimensional projective space, and V is linked to V ′ by a complete intersection X, then we say that V is minimally linked to V ′ if X is a codimension c complete intersection of smallest degree containing V . Gaeta showed that if V is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of...

‎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 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...

R.G. and Perez proved that under certain conditions the test ideal of a module closure agrees with trace closure. We use this fact to compute ideals various rings respect closures coming from their indecomposable maximal Cohen–Macaulay modules. also give an easier way hypersurface ring in three variables particular type matrix factorization.