A family A of sets is t-intersecting if the size of the intersection of every pair of sets in A is at least t, and it is an r-family if every set in A has size r. A wellknown theorem of Erdős, Ko, and Rado bounds the size of a t-intersecting r-family of subsets of an n-element set, or equivalently of (r−1)-dimensional faces of a simplex with n vertices. As a generalization of the Erdős-Ko-Rado ...

Strains of staphylococci prevalent in hospitals have been characterized by antibiotic-sensitivity patterns, phage type, serotype, and other criteria. Recently, T. Hofstad and P. Oeding (Acta Pathol. Microbiol. Scand. Suppl. 154, p. 311, 1962) reported on the development of a factor serum, 263-1, which agglutinated phage group I strains predominantly. This note is a corroboration of the findings...

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...

‎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...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.