The unit graph of a left Artinian ring

Let R be a ring with non-zero identity and U(R) be the group of units of R. The unit graph of R, denoted by G(R), is a graph defined on the elements of R, and two distinct vertices r and s are adjacent if and only if r+s∈U(R). We investigate connectivity, diameter and the girth of the unit graph of a left Artinian ring. Also, by providing an algorithm, we determine when the unit graph of a finite ring is Hamiltonian.     

Some Properties of a Cayley Graph of a Commutative Ring

Let R be a commutative ring with unity and R ⁺, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ?𝔸𝕐(R) and G _R , the Cayley graph Cay(R ⁺, Z*(R)) and the unitary Cayley graph Cay(R ⁺, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied G _R . In this article, we study ?𝔸𝕐(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ?𝔸𝕐(R). We also characterize all rings R whose ?𝔸𝕐(R) is planar. Moreover, we determine all finite rings R whose ?𝔸𝕐(R) is strongly regular. We prove that ?𝔸𝕐(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite rings R for which G _R is a strongly regular graph.     

Energy of unitary Cayley graphs and gcd-graphs

This work is based on ideas of Ili? [A. Ili?, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881-1889] on the energy of unitary Cayley graph. For a finite commutative ring R with unity , the unitary Cayley graph of R is the Cayley graph whose vertex set is R and the edge set is {{a,b}:a,b∈Randa-b∈R^×}, where R^× is the group of units of R. We study the eigenvalues of the unitary Cayley graph of a finite commutative ring and some gcd-graphs and compute their energy. Moreover, we obtain the energy for the complement of unitary Cayley graphs.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

Invertibility of Matrices Over Subrings

Given rings R ? S, consider the division closure 𝒟(R, S) and the rational closure ?(R, S) of R in S. If S is commutative, then 𝒟(R, S) = ?(R, S) = RT ^?1, where T = {t ∈ R | t ^?1 ∈ S}. We show that this is also true if we assume only that R is commutative.     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ? V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.     

Cryptogroups with an associate subgroup

A cryptogroup is a completely regular semigroup S on which Green’s relation $\mathcal{H}$ is a congruence. For a,x∈S, x is an associate of a if a=axa. A subgroup G of S is an associate subgroup of S if it contains precisely one associate of each element of S. Further, S is a regular (respectively normal) cryptogroup if $S/\mathcal{H}$ is a regular (respectively normal) band. We provide a construction of a general (respectively regular or normal) cryptogroup in terms of groups and functions. On this model of S, we find several conditions equivalent to S containing an associate subgroup G. We characterize several varieties of completely regular semigroups, provided with the unary operation s?s ^?, where s ^? is the associate of s in G. They include completely regular semigroups, (regular, normal) cryptogroups, completely simple semigroups, and their monoid and/or overabelian members.     

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)?{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.     

Zero-Divisor Graph with Respect to an Ideal

ABSTRACT

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ _I (R), is the graph whose vertices are the set {x ? R\I | xy ? I for some y ? R\I} with distinct vertices x and y adjacent if and only if xy ? I. In the case I = 0, Γ₀(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ _I (R) ? Γ _J (S) and Γ(R/I) ? Γ(S/J). We also discuss when Γ _I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ _I (R).     

Commutative Rings Whose Cozero-Divisor Graphs are Unicyclic or of Bounded Degree

Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ? Rb and b ? Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ? ?₂ × ?₄, ?₃ × ?₃, ?₂ × ?₂[x]/(x ²). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ? ?₂ × ?₄, ?₃ × ?₃, ?₂ × ?₂[x]/(x ²), ?₂[x, y]/(x, y)², ?₄[x]/(2x, x ²). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′(R)) ∈ {3, 4, ∞}.     

On Combinatorial Properties of Bands

In this paper we study the D-saturated property of bands defined in terms of their Cayley graphs Cay(S,C), where S is a band and C ? Z(S), the center of S. Also we characterize the Cayley graphs of bands. More generally, for a finite graph Γ =Cay(T, D), where T is a band and D ? Z(T), we give an algorithm for finding all bands S and C ? Z(S) such that Γ =Cay(S, C).     

Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case

It was proved by Glover and Maru?i? (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G=〈a,x∣a ²=x ^s =(ax)³=1,…〉 having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

Pseudo-amenability of certain semigroup algebras

In this paper, we investigate the pseudo-amenability of semigroup algebra ? ¹(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of ? ¹(S) is equivalent to the amenability of G.     

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.     

The Relative Brauer Group of an Affine Double Plane

We study algebra classes and divisor classes on a normal affine surface of the form z ² = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z ² ? f), and if R = k[x, y][f ^?1] and S = R[z]/(z ² ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H¹(G, Cl(T)) → H¹(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H¹(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples.