Narrow Results

To prove Kaplansky’s zero-divisor conjecture over ${𝔽}_2$, it suffices to prove it for a certain class of finitely presented groups where the relations are given by a pairing of generators. We associate Mealy automata to such pairings, and prove that the zero-divisor conjecture holds for groups corresponding to invertible automata with three states. In particular, there cannot be zero-divisors arising from an invertible pairing such that one of them has exactly three elements in its support.

Let $R$ be a commutative ring with identity and $I$ be an ideal of $R$. The zero-divisor graph of $R$ with respect to $I$, denoted by ${\mathrm{\Gamma }}_I(R)$, is the graph whose vertices are the set $\{x\in R\setminus I|xy\in I\mathrm{\text{for}}\mathrm{\text{some}}y\in R\setminus I\}$ with distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. The cozero-divisor graph ${\mathrm{\Gamma }}^{\prime }(R)$ of $R$ is the graph whose vertices are precisely the non-zero, non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x\notin yR$ and $y\notin xR$. In this paper, we introduced and investigated a new generalization of the cozero-divisor graph ${\mathrm{\Gamma }}^{\prime }(R)$ of $R$ denoted by ${\mathrm{\Gamma }}_I^{″}(R)$. In fact, ${\mathrm{\Gamma }}_I^{″}(R)$ is a dual notion of ${\mathrm{\Gamma }}_I(R)$.

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are nontrivial coverings over nice base quandles admit infinitely many nontrivial idempotents, and give their complete description. We show that the set of all these idempotents forms a quandle in itself. As an application, we deduce that the quandle ring of the knot quandle of a nontrivial long knot admit nontrivial idempotents. We consider free products of quandles and prove that integral quandle rings of free quandles have only trivial idempotents, giving an infinite family of quandles with this property. We also give a description of idempotents in quandle rings of unions and certain twisted unions of quandles.

In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle.

An element a of a ring R is called a weak zero-divisor if there exists a nonzero x of R such that xa is nilpotent. As a new generalization of nil clean rings, we define a ring R to be weak zero-clean if each of its elements is a sum of a weak zero-divisor and an idempotent. We give some examples and properties of weak zero-clean rings and discuss the relationships between weak zero-clean rings and related rings. Some extensions of weak zero-clean rings are studied. Also, we define and study uniquely weak zero-clean rings and give some characterizations of such rings.

We introduce the set S(R) of "strong zero-divisors" in a ring R and prove that: if S(R) is finite, then R is either finite or a prime ring. When certain sets of ideals have ACC or DCC, we show that either S(R) = R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. This is a finite union when R is a Noetherian ring. For a ring R with |S(R)| = p, a prime number, we characterize R for S(R) to be an ideal. Moreover R is completely characterized when R is a ring with identity and S(R) is an ideal with p² elements. We then consider rings R for which S(R)= Z(R), the set of zero-divisors, and determine strong zero-divisors of matrix rings over commutative rings with identity.

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.

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R₁ × R₂ × ⋯ × R_n, where each R_i is local and this representation of R is unique up to factors R_i with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

This paper is concerned with investigating the center and radius of zero-divisor and ideal-divisor graphs of finite commutative rings. In particular, we examine the interplay between the center and cut-sets of zero-divisor graphs. Of interest are conditions that need to be met in order for either of these sets to contain the other. We then explore the radius and central sets of ideal-divisor graphs, particularly the classification of the center via radius results.

In this paper, we will extend the notion of zero-divisor graph of commutative rings to zero-divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero-divisor graph of almost all abelian groups. Just a few classes of reduced abelian groups remain untouched in this paper.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc(G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc(G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc(G) to |V(G)| and show that there is a finite connected graph G with loc(G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group.

A ring $R$ is called eversible if every left zero-divisor in $R$ is also a right zero-divisor and conversely. This class of rings is a natural generalization of reversible rings. It is shown that every eversible ring is directly finite, and a von Neumann regular ring is directly finite if and only if it is eversible. We give several examples of some important classes of rings (such as local, abelian) that are not eversible. We prove that $R$ is eversible if and only if its upper triangular matrix ring $T_n(R)$ is eversible, and if $M_n(R)$ is eversible then $R$ is eversible.

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{\text{ann}}}_R(xy)\ne {\mathrm{\text{ann}}}_R(x)\cup {\mathrm{\text{ann}}}_R(y)$. In this paper, we study the metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

We prove that the zero-divisor graph of a direct product of matrices over finite zero-divisor free semirings uniquely determines the sizes of matrices and cardinalities of semirings in question. We also give an example that the semirings themselves are not necessarily uniquely determined.