Search

Let R be a ring with identity. The unit (respectively, unitary Cayley) graph of R is a simple graph G(R) (respectively, Γ(R)) with vertex set R, where two distinct vertices x and y are adjacent if and only if x + y (respectively, x − y) is a unit of R. In this paper, we explore when G(R) and Γ(R) are isomorphic for a finite ring R. Among other results, we prove that G(R) is isomorphic to Γ(R) for a finite ring R if and only if char(R/J(R))=2 or R/J(R)≅ℤ2 × S, where J(R) is the Jacobson radical of R and S is a finite ring. [ABSTRACT FROM AUTHOR]

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property. [ABSTRACT FROM AUTHOR]

The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring. A simple graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this note, we completely characterize the semipotent rings whose unit graphs are planar. As a consequence, we list all semilocal rings with planar unit graphs. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unitary Cayley graph of R is the simple graph with vertex set R , where two distinct vertices x and y are linked by an edge if and only if x − y is a unit of R. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]