We prove that S ⊆ {0, 1} is of second category if and only if for each f : ω → ⋃ n∈ω{0, 1} n there exists a sequence {an}n∈ω belonging to S such that for infinitely many i ∈ ω the infinite sequence {ai+n}n∈ω extends the finite sequence f(i). Let M denote the ideal of first category subsets of R. Let M({0, 1}) denote the ideal of first category subsets of the Cantor discontinuum {0, 1}. Obviousl...

Let O be the ring of integers of a totally real number field E and set G := ResE/Q(GL2). Fix an ideal c ⊂ O. For each ideal m ⊂ O let T (m) denote the mth Hecke operator associated to the standard compact open subgroup U0(c) of G(Af ). Setting X0(c) := G(Q)\G(A)/K∞U0(c), where K∞ is a certain subgroup of G(R), we use T (m) to define a Hecke cycle Z(m) ∈ IH2[E:Q](X0(c)×X0(c)). Here IH• denotes i...

notions of strongly regular, regular and left(right) regular $gamma$−semigroupsare introduced. equivalent conditions are obtained through fuzzy notion for a$gamma$−semigroup to be either strongly regular or regular or left regular.

The Frohman Kania-Bartoszynska ideal is an invariant associated to a 3-manifold with boundary and a prime p ≥ 5. We give some estimates of this ideal. We also calculate this invariant for some 3-manifolds constructed by doing surgery on a knot in the complement of another knot. Let p = 2d+ 1 ≥ 5 be a prime and let O = { Z[ζp] if p ≡ −1 (mod 4) , Z[ζp, i] = Z[ζ4p] if p ≡ 1 (mod 4) . If M is an o...

Let l and m be the ideals associated with Laver and Miller forcing, respectively. We show that add(l) < cov(l) and add(m) < cov(m) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal ≤ h. Introduction and Notation In this paper we investigate the ideals connected with the classical tree forcings introduced by Laver [La] and Miller [Mi]. Laver for...

Let M denote the ideal of first category subsets of R. We prove that min{card X : X ⊆ R,X 6∈ M} is the smallest cardinality of a family S ⊆ {0, 1} with the property that for each f : ω −→ ⋃ n∈ω{0, 1} n there exists a sequence {an}n∈ω belonging to S such that for infinitely many i ∈ ω the infinite sequence {ai+n}n∈ω extends the finite sequence f(i). We inform that S ⊆ {0, 1} is not of first cate...

let $r$ be a commutative ring with identity and $mathbb{a}(r)$ be the set   of ideals of $r$ with non-zero annihilators. in this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $r$, denoted by $mathbb{ag}_p(r)$. it is a (undirected) graph with vertices $mathbb{a}_p(r)=mathbb{a}(r)cap mathbb{p}(r)setminus {(0)}$, where   $mathbb{p}(r)$ is...

Let (R,m) be a local ring, I a proper ideal of R and M a finitely generated R-module of dimension d. We discuss the local homology modules of H I (M). When M is Cohen-Macaulay, it is proved that H d m(M) is co-CohenMacaulay of N.dimension d and H x d (H m(M)) = M̂ where x = (x1, . . . , xd) is a system of parameters for M . 2000 Mathematics Subject Classification: 13C14, 13D45

1. Some basic facts Denote by O = ON+1 the ring of germs of holomorphic functions f = f(z0, · · · , zN ) defined in a neighborhood of ~0 ∈ CN+1. We denote by m ⊂ O the maximal ideal of O, f ∈ m ⇐⇒ f(~0) = 0. Let f ∈ m. Assume ~0 is an isolated critical point of f , i.e. ~0 is an isolated point of the variety ∂zif = 0, ∀i = 0, · · · , N. We define the Jacobian ideal of f to be the ideal Jf ⊂ O g...

Let l and m be the ideals associated with Laver and Miller forcing, respectively. We show that add(l) < cov(l) and add(m) < cov(m) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal ≤ h. Introduction and Notation In this paper we investigate the ideals connected with the classical tree forcings introduced by Laver [La] and Miller [Mi]. Laver for...