Narrow Results

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants W_{α, q}(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G.

This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

We calculate, only in terms of a commutative unital ring R of prime characteristic p and an abelian p-mixed group G, the classical Warfield q-invariants W_α,q(VR(G)) of the group VR(G) of all normalized units in the group ring R(G). This continues our results in (Extr. Math., 2005), (Collect. Math., 2008) and (J. Alg. Appl., 2008).

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. First, in this article we determine when the complement of the zero-divisor graph of R is connected and also determine its diameter when it is connected. Second, in this article we study the relationship between the connectedness of the complement of the zero-divisor graph of R to that of the connectedness of the complement of the zero-divisor graph of T where either T = R[x] or T = R[[x]] and we study the relationship between their diameters in the case when both the graphs are connected. Finally, we give some examples to illustrate some of the results proved in this article.

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam(𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr(𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with 𝔸(R) be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph 𝔸𝔾(R) that its vertices are 𝔸(R)* = 𝔸(R)\{(0)} in which for every distinct vertices I and J, I — J is an edge if and only if IJ = (0). First, we study the diameter of 𝔸𝔾(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(𝔸𝔾(R)) ≤ 2 or R is reduced and χ(𝔸𝔾(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(𝔸𝔾(R)) = cl(𝔸𝔾(R)). Moreover, if χ(𝔸𝔾(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(𝔸𝔾(R)) = cl(𝔸𝔾(R)) = n. Finally, we show that for a Noetherian ring R, cl(𝔸𝔾(R)) is finite if and only if for every ideal I of R with I² = (0), I has finite number of R-submodules.

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 Γ_S(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 Γ_S(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

Let R be a commutative ring and M be an R-module with a proper submodule N. The total graph of M with respect to N, denoted by T(Γ_N(M)), is investigated. The vertex set of this graph is M and for all x, y belonging to M, x is adjacent to y if and only if x + y ∈ M(N), where M(N) = {m ∈ M : rm ∈ N for some r ∈ R - (N : M)}. In this paper, in addition to studying some algebraic properties of M(N), we investigate some graph theoretic properties of two important subgraphs of T(Γ_N(M)) in the cases depending on whether or not M(N) is a submodule of M.

Let R be a commutative unital ring of arbitrary characteristic and let G be a multiplicative Abelian group. For the group ring RG we completely calculate the number (finite or infinite) of its idempotents only in terms of R, G and their sections. This strengthens our previous results in Sarajevo J. Math. (2011) and Filomat (2012).

To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395–1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213–228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771–778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125–1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with |R/I| < |R| for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.

In this paper we introduce and study the spectrum graph of a commutative ring R, denoted by 𝔸𝔾_s(R), that is, the graph whose vertices are all non-zero prime ideals of R with non-zero annihilator and two distinct vertices P₁, P₂ are adjacent if and only if P₁P₂ = (0). This is an induced subgraph of the annihilating-ideal graph 𝔸𝔾(R) of R. Among other results, we present the structures of all graphs which can be realized as the spectrum graph of a commutative ring. Then we show that for a non-domain Noetherian ring R, 𝔸𝔾_s(R), is a connected graph if and only if 𝔸𝔾_s(R) is a star graph if and only if 𝔸𝔾_s(R) ≅ K₁, K₂ or K_1,∞, where K_n is a complete graph with n vertices and K_1,∞ is a star graph with infinite vertices. Also, we completely characterize the spectrum graphs of Artinian rings. Finally, as an application, we present some relationships between the annihilating-ideal graph and its spectrum subgraph.

Let $R$ be a commutative ring with identity and $I^{\ast }(R),$ the set of all nontrivial proper ideals of $R$. The intersection graph of ideals of $R$, denoted by ${ℐ}(R)$, is a simple undirected graph with vertex set as the set $I^{\ast }(R)$, and, for any two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\ne (0)$. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring $R$, ${ℐ}(R)$ is a projective graph if and only if $R\cong R_1\times F_1,$ where $R_1$ is a local principal ideal ring with maximal ideal ${𝔪}_1$ of nilpotency three and $F_1$ is a field. Furthermore, it is shown that for an Artinian ring $R,$$\overline{\gamma }({ℐ}(R))=2$ if and only if $R\cong R_1\times R_2,$ where each $R_i(1\le i\le 2)$ is a local principal ideal ring with maximal ideal ${𝔪}_i\ne (0)$ such that ${𝔪}_i^2=(0).$

In this paper, we consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the R-module U does not exceed 1. We also describe the Geigle–Lenzing perpendicular subcategory $U^{{\perp }_{0,1}}$ in $R-Mod$. Assuming additionally that the ring U and all the rings $R/I$ are perfect, we show that all flat R-modules are U-strongly flat. Thus, we obtain a generalization of some results of the paper [6], where the case of the localization $U=S^{-1}R$ of the ring R at a multiplicative subset $S\subset R$ was considered.

Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T_Γ(R), is the (undirected) graph with vertices set R. For any two distinct elements x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we obtain certain fundamental properties of the total graph of ℤ_n × ℤ_m, where n and m are positive integers. We determine the clique number and independent number of the total graph T_Γ(ℤ_n × ℤ_m).

Let $R$ be a commutative ring with identity. In this paper, we consider a simple graph associated with $R$ denoted by ${\mathrm{\Omega }}_R^{\ast }$, whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $J\mathrm{\text{Ann}}(I)=(0)$ or $I\mathrm{\text{Ann}}(J)=(0)$. In this paper, we initiate the study of the graph ${\mathrm{\Omega }}_R^{\ast }$ and we investigate its properties. In particular, we show that ${\mathrm{\Omega }}_R^{\ast }$ is a connected graph with $\mathrm{\text{diam}}({\mathrm{\Omega }}_R^{\ast })\le 3$ unless $R$ is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings $R$ with at least two maximal ideals for which ${\mathrm{\Omega }}_R^{\ast }$ are planar.

Let $R$ be a commutative ring with identity and ${𝔘}_R$ be the set of all non-zero non-units of $R$. The co-annihilating graph of $R$, denoted by ${{𝒞}{𝒜}}_R$, is a graph with vertex set ${𝔘}_R$ and two vertices $a$ and $b$ are adjacent whenever $\mathrm{\text{Ann}}(a)\cap \mathrm{\text{Ann}}(b)=(0)$. In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

Let $R$ be a commutative ring with identity. The co-annihilating-ideal graph of $R$, denoted by ${{𝒜}}_R$, is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices $I$ and $J$ are adjacent whenever $\mathrm{\text{Ann}}(I)\cap \mathrm{\text{Ann}}(J)=\{0\}$. In this paper, we study the planarity and genus of ${{𝒜}}_R$. In particular, we characterize all Artinian rings $R$ for which the genus of ${{𝒜}}_R$ is zero or one.

The concept of associating a graph to a ring was initiated by Beck in 1988. The zero-divisor graph of a commutative ring $R$ with unity ($1\ne 0$) is defined as a simple graph, denoted by $\mathrm{\Gamma }(R)$, where elements of the ring $R$ represent vertices and two distinct vertices in $\mathrm{\Gamma }(R)$ are adjacent if the product of the corresponding elements is zero in $R$. Here, we extend the notion of zero-divisor graphs to signed graphs. We introduce four types of signing to the zero-divisor graphs. Further, we characterize rings for which these four types of signed graphs, its lined signed graphs and their negations are balanced, clusterable, sign-compatible, canonically sign-compatible, and canonically consistent.