Tongsuo Wu

On directed zero-divisor graphs of finite rings

For an artinian ring R, the directed zero-divisor graph @C(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in @C(R) with y^2=0, then |R|=4 and R={0,x,y,z}, where x and z are left identity elements and yx=0=yz. Such a ring R is also the only ring such that @C(R) has exactly one source. This shows that @C(R) cannot be a network for any finite or infinite ring R.

The Zero-divisor Graphs of Posets and an Application to Semigroups

In this paper, we introduce the notion of a compact graph. We show that a simple graph is a compact graph if and only if G is the zero-divisor graph of a poset, and give a new proof of the main result in Halaš and Jukl (Discrete Math 309:4584–4589, 2009) stating that if G is the zero-divisor graph of a poset, then the chromatic number and the clique number of G coincide under a mild assumption. We observe that the zero-divisor graphs of reduced commutative semigroups (rings) are compact, thus provide a large class of graphs G that could be realized as zero-divisor graphs of posets. In addition, using these results, we give some equivalent descriptions for the zero-divisor graphs of posets and reduced commutative semigroups with 0 respectively.

On bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r>=3.

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x2 = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of Kn if n + 1 = pq for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.

Finitely generated projective modules over exchange rings

This paper studies finitely generated projective modules over exchange rings. We prove that cancellation holds inp(R), andKo(R) is completely determined by the continuous maps from the spectrum ofR toZ ifR is an exchange ring andR/J(R) is a ring with central idempotent elements.

Zero-Divisor Semigroups and Some Simple Graphs

In this article, we study commutative zero-divisor semigroups determined by graphs. We prove that for all n ≥ 4, the complete graph K n together with two end vertices has a unique corresponding zero-divisor semigroup, while the complete graph K n together with three end vertices has no corresponding semigroups. We determine all the twenty zero-divisor semigroups whose zero-divisor graphs are the complete graph K 3 together with an end vertex.

Sub-semigroups determined by the zero-divisor graph

In this paper we study sub-semigroups of a finite or an infinite zero-divisor semigroup S determined by properties of the zero-divisor graph @C(S). We use these sub-semigroups to study the correspondence between zero-divisor semigroups and zero-divisor graphs. In particular, we discover a class of sub-semigroups of reduced semigroups and we study properties of sub-semigroups of finite or infinite semilattices with the least element. As an application, we provide a characterization of the graphs which are zero-divisor graphs of Boolean rings. We also study how local property of @C(S) affects global property of the semigroup S, and we discover some interesting applications. In particular, we find that no finite or infinite two-star graph has a corresponding nil semigroup.

On the stable range condition of exchange rings

The purpose of this paper is to prove the following facts:An exchange ring R has stable range at most n if and only if for any module isomorphism Rn ⊕ A ≅ R⊕B, A is isomorphic to a direct summand of B, if and only if for any regular element x of Rn, x = xux for some unimodular column u of nR. For any module M with the finite exchange property, the stable range of EndR(M) is at most n if and only if M satisfies the n-weak cancellation. These results generalize a result of Menal and Moncasi [5] and a recent result of Camillo and Yu [2].

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use

The power-substitution condition of endomorphism rings of quasi-projective modules

\mathscr{C}(R)