The Compressed Annihilator Graph of a Commutative Ring

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AG_E(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ? ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.     

Classification of finite commutative rings with planar,toroidal, and projective line graphs associated with Jacobson graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J _R , is a graph with vertex-set R J(R), such that two distinct vertices a and b in R J(R) are adjacent if and only if 1 ? ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L(J _R ). In this paper, we characterize all finite commutative rings R such that the graphs L(J _R ) are planar, toroidal or projective.     

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with \(1\ne 0\) and the additive group \(R^+\). Several graphs on R have been introduced by many authors, among zero-divisor graph \(\Gamma _1(R)\), co-maximal graph \(\Gamma _2(R)\), annihilator graph AG(R), total graph \( T(\Gamma (R))\), cozero-divisors graph \(\Gamma _\mathrm{c}(R)\), equivalence classes graph \(\Gamma _\mathrm{E}(R)\) and the Cayley graph \(\mathrm{Cay}(R^+ ,Z^*(R))\). Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to \(\mathrm{Cay}(R^+ ,Z^*(R))\). Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with \(|\mathrm{Max}(R)|=n \ge 3\), \( \Gamma _1(R) \simeq \Gamma _2(R)\) if and only if \(R\simeq \mathbb {Z}^n_2\); if and only if \(\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)\). Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.     

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by J_R, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 ? xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S_n. In this paper, we investigate the genus number of the compact Riemann surface in which J_R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that J_R is toroidal.     

Commutative rings with genus two annihilator graphs

Let R be a commutative ring and Z(R)* be its set of all nonzero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by Badawi [8 Badawi, A. (2014). On the annihilator graph of a commutative ring. Commun. Algebra 42(1):108–121.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In this paper, we classify the finite commutative rings whose AG(R) are projective. Also we determine all isomorphism classes of finite commutative rings with identity whose AG(R) has genus two.     

Eulerian graphs and automorphisms of a maximal graph

Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ? R ₁×R ₂× · · · ×R _n, where the R _i is a local ring of cardinality p _i ^αi for all i, and the p _i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R.     

Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

Torsion-free modules with UA-rings of endomorphisms

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring End _R (A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.     

On weakly nil-clean rings

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring \(\mathbb{M}_n (R)\) is weakly nil-clean, and to show that the endomorphism ring End _D (V) over a vector space V _D is weakly nil-clean if and only if it is nil-clean or dim(V) = 1 with D?= ?₃.     

Every graph is (2,3)-choosable

A total weighting of a graph G is a mapping ? that assigns to each element z ∈ V (G)∪E(G) a weight ?(z). A total weighting ? is proper if for any two adjacent vertices u and v, ∑ _e∈E(u) ?(e)+?(u)≠∑ _e∈E(v) ?(e)+?(v). This paper proves that if each edge e is given a set L(e) of 3 permissible weights, and each vertex v is given a set L(v) of 2 permissible weights, then G has a proper total weighting ? with ?(z) ∈ L(z) for each element z ∈ V (G)∪E(G).     

On the total multiplicity of zeros of generalized polynomials related to nonoscillating trajectories

For a continuous curve L = {x: x = Z(t), t ∈ [a, b]} in R ⁿ , we study the number of zeros of the function l _h (t) = 〈h, Z(t)〉, where h ∈ R ⁿ . We introduce the notion of multiple zeros for such functions and study the possibility of estimating the total multiplicity of such zeros under the assumption that the system {z ₁(t), z ₂(t), …, z _n (t)} of coordinates of the function Z(t) is a Chebyshev system on [a, b].     

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x ²) = 2xF(x) + x ² w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

The regular digraph of ideals of a commutative ring

Let R be a commutative ring and Max?(R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by \(\overrightarrow{\Gamma_{\mathrm{reg}}}(R)\), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γ_reg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max?(R)|?1≦ω(Γ_reg(R))≦|Max?(R)| and \(\chi(\Gamma_{\mathrm{ reg}}(R)) = 2|\mathrm{Max}\, (R)| -k-1\), where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that \(\overrightarrow{\Gamma_{\mathrm{ reg}}}(R)\) is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.     

Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x ₁,..., x _n ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r ₁,..., r _n ): r _i ∈ R} be the set of all evaluations of f(x ₁,..., x _n ) in R, while A = {[G (f(r ₁,..., r _n )), f(r ₁,..., r _n )]: r _i ∈ R}, and let C _R (A) be the centralizer of A in R; i.e., C _R (A) = {a ∈ R: [a, x] = 0, ? _x ∈ A }. We prove that if A ≠ (0), then C _R (A) = Z(R).     

Fekete-Szegö problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Given α ∈ [0, 1], let h _α (z):= z/(1 - αz), z ∈ D:= {z ∈ D: |z| < 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to h _α if there exists δ ∈ (-π/2, π/2) such that Re{e^iδ zf′(z)/h _α (z)} > 0, z ∈ D. For the class ? (h _α ) of all close-to-convex functions with respect to h _α , the Fekete-Szegö problem is studied.     

Annihilating and power-commuting generalized skew derivations on lie ideals in prime rings

Let R be a prime ring of characteristic different from 2 and 3, Q_r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y ∈ R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(x)D(y) for all x, y ∈ R.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

Some Properties of the Idempotent Graph of a Ring

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam\({(I(M_n(D))) = 4}\), for all natural number \({n \geq 4}\) and diam\({(I(M_3(D))) = 5}\), where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied.