zero-divisor

In this paper, we introduce the zero-divisor associate graph $\Gamma_D(R)$ over a finite commutative ring $R$. It is a simple undirected graph whose vertex set consists of all non-zero elements of $R$, and two vertices $a, b$ are adjacent if and only if there exist non-zero zero-divisors $z_1, z_2$ in $R$ such that $az_1=bz_2$. We determine the necessary and sufficient conditions for connectedness and completeness of $\Gamma_D(R)$ for a unitary commutative ring $R$. The chromatic number of $\Gamma_D(R)$ is also studied. Next, we characterize the rings $R$ for which $\Gamma_D(R)$ becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as $\Gamma_D(R)$, characterizing the associated ring $R$ in each case.

For an R-module M and f ∈ M∗ = Hom(M, R), let Zf(M) and Regf(M) be the sets of all zero-divisors elements and regular elements of M with re- spect to f, respectively. In this paper, we introduce the total graph of M with respect to f, denoted by T(Γf(M)), which is the graph with all the elements M as vertices, and for distinct elements m, n ∈ M, m and n are adjacent and only if m + n ∈ Zf(M). We also study the subgraphs Z(Γf(M)) and Reg(Γf(M)) with vertices Zf(M) and Regf(M), respectively.

Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, we obtain several results on this probability including a computing formula and characterizations. In particular, the computing formula for the $P(\mathbb{F}_qG)$ are established where $G$ is the cyclic group $C_n$, the Quaternion group $Q_8$, the symmetric group $S_3$ and $F_q$ is a finite field.

Let $R$ be a non trivial finite commutative ring with identity and $G$ be a non trivial group‎. ‎We denote by $P(RG)$ the probability that the product of two randomly chosen elements of a finite group ring $RG$ is zero‎. ‎We show that $P(RG)<\frac{1}{4}$ if and only if $RG\ncong \mathbb{Z}_2C_2,\mathbb{Z}_3C_2‎, ‎\mathbb{Z}_2C_3$‎. ‎Furthermore‎, ‎we give the upper bound and lower bound for $P(RG)$‎. ‎In particular‎, ‎we present the general formula for $P(RG)$‎, ‎where $R$ is a finite field of characteristic $p$ and $|G|\leq 4$‎.

Let $R$ be a commutative ring with unity not equal to zero and let $Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) subseteq V(G)$ such that $|N(v) cap C(G)| = 1$ for all $v in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. We also show that if $Gamma(R)$ is a regular graph on $|Z^*(R)|$ number of vertices, then $R$ is a reduced ring and $|Z^*(R)| equiv 0 (mod ~2)$, where $Z^*(R)$ is a set of non-zero zero-divisors of $R$. We provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. Finally, we determine the cardinality of a total perfect code in $Gamma(R)$ and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity.

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This paper introduces and studies the notion of Property (A) of a ring R or an R-module M along an ideal I of R. For instance, any module M over R satisfying the Property (A) do satisfy the Property (A) along any ideal I of R. We are also interested in ideals I which are A-module along themselves. In particular, we prove that if I is contained in the nilradical of R, then any R-module is an A-module along I and, thus, I is an A-module along itself. Also, we present an example of a ring R possessing an ideal I which is an A-module along itself while I is not an A-module. Moreover, we totally characterize rings R satisfying the Property (A) along an ideal I in both cases where I⊆\Z(R) and where I⊈\Z(R). Finally, we investigate the behavior of the Property (A) along an ideal with respect to direct products.

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in R[[X]], then it will holds in R.

Let $A$ be a commutative ring with nonzero identity, and $1leq n<infty$ be an integer, and $R=Atimes Atimescdotstimes A$ ($n$ times). The total dot product graph of $R$ is the (undirected) graph $TD(R)$ with vertices $R^*=Rsetminus {(0,0,dots,0)}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xcdot y=0in A$ (where $xcdot y$ denote the normal dot product of $x$ and $y$).  Let $Z(R)$ denote the set of all zero-divisors of $R$.  Then the zero-divisor dot product graph of $R$ is the induced subgraph $ZD(R)$ of $TD(R)$ with vertices  $Z(R)^*=Z(R)setminus {(0,0,dots,0)}$. It follows that if  $Gamma(A)$ is not  perfect, then  $ZD(R)$ (and hence $TD(R)$) is not  perfect.In this paper we investigate perfectness of the graphs $TD(R)$ and $ZD(R)$.

Let Γ=(V,E) be a graph and ‎W_(‎a)={w_1,…,w_k } be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v,w_1),‎…‎ ,a_Γ (v,w_k)) is the adjacency representation of v with respect to W in which a_Γ (v,w_i )=min{2,d_Γ (v,w_i )} and d_Γ (v,w_i ) is the distance between v and w_i in Γ. W_a is called as an adjacency resolving set for Γ if distinct vertices of Γ have distinct adjacency representations w.r.t W_a. The size of the smallest adjacency resolving set is the adjacency metric dimension of Γ and is denoted by ‎dim_a‎(Γ). In this paper, we prove that dim_a(Γ_E (Z_(P^n ) ))=⌈(n-2)/2⌉. Also, we show that Γ_E (Z_(p^2n ) )≅Γ_E (R/I) in which p is a prime number, n is a natural number and I is a 2-absorbing ideal of the ring R which has a minimal primitive decomposition in the form of the intersection of n primitive ideals. Finally we conclude that dim_a⁡〖(Γ_E (R/I))=n-1〗.

A positive solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of sufficiently large odd periods n>10 10 n>1010 obtained previously by S. V. Ivanov and R. Mikhailov extended to all odd periods n≥665 n≥665 ý.