Robert Peters,1,* Shintaro Hoshino,2 Norio Kawakami,1 Junya Otsuki,2,3 and Yoshio Kuramoto2 1Department of Physics, Kyoto University, Kyoto 606-8502, Japan 2Department of Physics, Tohoku University, Sendai 980-8578, Japan 3Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany (Received 21 February 2013;...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...

Let be a commutative Noetherian ring and let I be a proper ideal of . D’Anna and Fontana in [6] introduced a new construction of ring,  named amalgamated duplication of along I. In this paper by considering the ring homomorphism , it is shown that if , then , also it is proved that if , then there exists  such that . Using this result it is shown that if is generically Cohen-Macaulay (resp. gen...

The combinatorial properties of Cohen forcing imply the existence of a countably closed, א2-c.c. forcing notion P which adds a C(ω2)-name Q for a σ-centered poset such that forcing with Q over V P×C(ω2) adds a real not split by V C(ω2) ∩ [ω] and preserves that all subfamilies of size ω1 of the Cohen reals are unbounded.

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...