THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS I

On the coloring of the annihilating-ideal graph of a commutative ring

The polynomial ring over a Goldie ring need not be a Goldie ring

In this paper, we study ace (ascending chain condition) and dec (descending chain condition) on different types of subgroups of LCA (locally compact abelian) groups, such as open subgroups, compact subgroups, discrete subgroups, metrizable subgroups, closed divisible subgroups, proper dense subgroups. We characterize compactly generated LCA groups as the class of LCA groups whose open subgroups satisfy ace; the compactly cogenerated groups as the class of LCA groups whose discrete subgroups satisfy dec. We also show that ace and dec on the following classes of subgroups are pairwise equivalent: (a) closed subgroups (b) closed totally disconnected subgroups (c) closed (7-compact subgroups (d) closed metrizable subgroups. We also obtain a characterizati on of those LCA groups which contain no proper dense subgroups.

A nearring N is weakly divisible (wd-nearring) if, for each x, y ∈ N, there exists an element z ∈ N such that xz = y or yz = x. In the first part the structure of wd-nearrings with ascending or descending chain conditions is studied and wd-nearrings with ascending chain condition on the right annihilators are completely described. In the second part all zerosymmetric wd-nearrings on the group G of integers (mod p n ), p prime, in which pG is the set of all the nilpotent elements are characterised and constructed. At the end the possibility to construct partially balanced block designs from finite wd-nearrings is investigated. Non balanced designs and their related association schemes are derived from wd-nearrings having cyclic additive group G of order 2 n . For wd-nearrings on a cyclic additive group of order p n , p prime and p ≠ 2, an example is given.

Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.

Let R be a commutative ring with identity and $$\mathbb {A}(R)$$ be the set of all annihilator ideals of R. The annihilator-ideal graph of R, denoted by $$A_I(R)$$ , is a simple graph with the vertex set $$\mathbb {A}^*(R) := \mathbb {A}(R)\setminus \{(0)\}$$ , and two distinct vertices I and J are adjacent if and only if $$ann(IJ)\ne ann(I) \cup ann(J)$$ . In this paper, we characterize all isomorphism classes of Artinian commutative rings whose $$A_I(R)$$ has genus one and crosscap one.

Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors. The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this article, we characterize, up to isomorphism, all of finite commutative rings whose total graphs have vertex-arboricity (arboricity) two or three. Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if [Formula: see text]. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs.

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam (ITΓ(R)) ≤ 2 and gr (ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all commutative Artin rings R for which ITΓ(R) is either planar or toroidal or genus two.

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

Let$R$be a commutative ring. The regular digraph of ideals of$R$, denoted by$\Gamma (R)$, is a digraph whose vertex set is the set of all nontrivial ideals of$R$and, for every two distinct vertices$I$and$J$, there is an arc from$I$to$J$whenever$I$contains a nonzero divisor on$J$. In this paper, we study the connectedness of$\Gamma (R)$. We also completely characterise the diameter of this graph and determine the number of edges in$\Gamma (R)$, whenever$R$is a finite direct product of fields. Among other things, we prove that$R$has a finite number of ideals if and only if$\mathrm {N}_{\Gamma (R)}(I)$is finite, for all vertices$I$in$\Gamma (R)$, where$\mathrm {N}_{\Gamma (R)}(I)$is the set of all adjacent vertices to$I$in$\Gamma (R)$.

The classical structure theorems of algebraic systems usually assume some type of finiteness condition. The most common finiteness restriction is a chain condition. Thus the proofs of the fundamental structure and decomposition theorems for lattices have customarily required the ascending chain condition. Moreover, these theorems generally fail to hold in arbitrary lattices. Nevertheless, there are important examples of decomposition theorems for lattices associated with abelian groups and rings in which the ascending chain condition does not hold. These lattices have in common another distinctive property—they are compactly generated. Namely, the lattice is generated by a collection of elements which are finitely dependent in the sense that any such element is contained in the union of a set of lattice elements if and only if it is contained in the union of a finite subset. The compact elements of the lattice of ideals of a ring are the finitely generated ideals. Likewise the compact elements of a lattice of congruence relations are the congruence relations generated by collapsing a finite collection of element pairs. More generally, the lattice of congruence relations and the lattice of subsystems of a universal algebra are compactly generated. Since structure theorems for an algebraic system correspond to decomposition theorems in the lattice of congruence relations, this strongly suggests that compactly generated lattices are the appropriate domain in which to study decomposition theory. Furthermore, since every lattice satisfying the ascending chain condition is trivially compactly generated, it follows that the classical case will be subsumed in the more general theory.

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

Let R be a commutative ring with non-zero identity and Z^∗(R) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z^∗(R) are adjacent if and only if xy = 0. In this paper, the eigenvalues of Γ(Zn) for n = p^2q^2, where p and q are distinct primes, are investigated. Also, the girth, diameter, clique number and stability number of this graph are found.

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of ΓS(R). Moreover, we will improve and generalize some results for the total and the unit graphs.