The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread.

In this note, we introduce the notions of fuzzy characteristic and fuzzy divisor of zero of a ring, and investigate some properties of these notions in connection with fuzzy ideals of a ring. AMS Mathematics Subject Classification (2000): 04A72, 2N25

We review some methods of associating graphs to rings. The main emphasis is on zero-divisor graphs and comaximal ones. Some new results in both directions are presented. AMS Mathematics Subject Classification (2000): 13A99, 05C99, 16S50

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.

The vertex-connectivity and edge-connectivity of the zero-divisor graph associated to a finite commutative ring are studied. It is shown that the edgeconnectivity of ΓR always coincides with the minimum degree. When R is not local, it is shown that the vertex-connectivity also equals the minimum degree, and when R is local, various upper and lower bounds are given for the vertex-connectivity.

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 ΓI(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 literat...

Continuing arguments presented [1] or announced [2, 3] here, zerodivisor (ZD) foundations for scale-free networks (evinced, in particular, in the “fractality” of the Internet) are decentralized. Spandrels, quartets of ZD-free or “hidden” box-kite-like structures (HBKs) in the 2N+1ions, are “exploded” from (and uniquely linked to) each standard boxkite in the 2N-ions, N ¥ 4. Any HBK houses, in a...

We consider the notion of a Φ-zero divisor, where Φ is Young function, on discrete group. extend some results about p-divisors free groups by Linnell and Puls to divisors.

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and the k-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. Let R be a com...