2. Primary ideals in C(X). C(X) denotes the ring of real-valued continuous functions on the topological space X, under ordinary pointwise addition and multiplication. By a primary ideal we mean an ideal which is contained in at most one maximal ideal. An O-ideal is an ideal / such that for each pair/i,/2G/ there exists eEI, depending on/i and f2, such that/<«=/.-, for i = l, 2 [7; 4]. A prime-l...

In this paper, we first introduce the new concept of completely semiprime fuzzy ideals of an ordered semigroup S, which is an extension of completely semiprime ideals of ordered semigroup S, and investigate some its related properties. Especially, we characterize an ordered semigroup that is a semilattice of simple ordered semigroups in terms of completely semiprime fuzzy ideals of ordered semi...

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

Recently Swatee Naik and Theodore Stanford proved that two Sequivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to ordered links. We define a new algebraic relation on Seifert matrices, called Strong S-equivalence, and prove that two oriented, ordered links L and L are rela...

We performed a large-scale crawl of the World Wide Web, covering 6.9 Million domains, including all high-traffic sites of the Internet. We present a study of the correlations found between quantities measuring the structural relevance of each node in the network (the inand out-degree, the local clustering coefficient, the first-neighbor in-degree and the Alexa rank). We find that some of these ...

the minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of c(x). the necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) i is an intersection of o , s, when x is basically disconnected. (iii) i=o , when x x ha...

This article concerns commutative factor rings for ideals contained in the center. A ring $R$ is called CIFC if $R/I$ some proper ideal $I$ of with $I\subseteq Z(R)$, where $Z(R)$ center $R$. We prove that (i) a $R$, $W(R)$ contains all nilpotent elements (hence Köthe's conjecture holds $R$) and $R/W(R)$ reduced ring; (ii) strongly bounded $R/N_*(R)$ $0\neq N_*(R)\subseteq (resp., $N_*(R)$) Wed...