The Intersection Theorem for Direct Products

The Intersection Theorem for Direct Products

Extension of Krull's intersection theorem for fuzzy module

‎In this article we introduce $mu$-filtered fuzzy module with a family of fuzzy submodules.  It shows the relation between $mu$-filtered fuzzy modules and crisp filtered modules by level sets. We investigate fuzzy topology on the $mu$-filtered fuzzy module and apply that to introduce fuzzy completion. Finally we extend Krull's intersection theorem of fuzzy ideals by using concept $mu$-adic comp...

An Asymptotic Complete Intersection Theorem for Chain Products

For positive integers k, ` and n let N`(n, k) = {a = (a1, . . . , an) : ai ∈ {0, 1, . . . , k}, i = 1, . . . , n,∑ ai = `}. A family F ⊆ N`(n, k) is called t-intersecting if for all a, b ∈ F there exist t coordinates i1, . . . , it such that ai j , bi j ≥ 1 holds for j = 1, . . . , t . Define M`(n, k, t) = max{|F | : F ⊆ N`(n, k),F is t-intersecting}. N`(n, k) can be viewed as the `-th level of...

The Krull Intersection Theorem

Let R be a commutative ring, / an ideal in R, and A an i?-module. We always have 0 £= 0 £ I(\~=1 I A £ f|ϊ=i JM. where S is the multiplicatively closed set {1 — i\ie 1} and 0 = 0s Π A = {α G A13S 6 S 3 sα = 0}. It is of interest to know when some containment can be replaced by equality. The Krull intersection theorem states that for R Noetherian and A finitely generated I Π*=i I A = Π~=i IA. Si...

An Erdos-Ko-Rado Theorem for Direct Products

Let n i , k i be positive integers , i 5 1 , . . . , d , satisfying n i > 2 k i . Let X 1 , . . . , X d be pairwise disjoint sets with u X i u 5 n i . Let * be the family of those ( k 1 1 ? ? ? 1 k d )-element sets which have exactly k i elements in X i , i 5 1 , . . . , d . It is shown that if ̂ ’ * is an intersecting family then u ̂ u / u * u < max i k i / n i , and this is best possible . The ...