For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undi...

We introduce a graph structure \(\Gamma^{\ast}_2(R)\) for commutative rings with unity. study some of the properties \(\Gamma^{\ast}_2(R)\). Also we parameters and find which is split.

let $r$ be a commutative ring with identity‎. ‎we use‎ ‎$varphi (r)$ to denote the comaximal ideal graph‎. ‎the vertices‎ ‎of $varphi (r)$ are proper ideals of r which are not contained‎ ‎in the jacobson radical of $r$‎, ‎and two vertices $i$ and $j$ are‎ ‎adjacent if and only if $i‎ + ‎j = r$‎. ‎in this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎assoc...

The intersection ideal graph Γ(S) of a semigroup S is simple undirected whose vertices are all nontrivial left ideals and two distinct I, J adjacent if only their nontrivial. In this paper, we investigate the connectedness Γ(S). We show that connected, then diameter at most two. Further, classify semigroups in terms such obtain domination number, independence girth strong metric dimension have ...

We classify the gauge-invariant ideals in the C∗-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C∗-algebras of arbitrary infinite graphs.

  ‎The rings considered in this article are commutative with identity which admit at least two maximal ideals‎.  ‎This article is inspired by the work done on the comaximal ideal graph of a commutative ring‎. ‎Let R be a ring‎.  ‎We associate an undirected graph to R denoted by mathcal{G}(R)‎,  ‎whose vertex set is the set of all proper ideals I of R such that Inotsubseteq J(R)‎, ‎where J(R) is...