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Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if RZD(R) is finite, then R is finite  

Let G = {g1,...,gn} be a finite abelian group. Consider the complete graph with the vertex set {g1.....,.....g n}. The G-coloring of Kn is a proper edge coloring in which the color of edge {gi,gj} gi g i + gj, 1 ≤ i < 3 ≤ n. We prove that in the G-coloring of the complete graph Kn, there exists a multicolored Hamilton path if G is not an elementary abelian 2-group. Furthermore, we show that if n is odd, then the G-coloring of Kn can be decomposed into multicolored 2-factors and there are exactly lr/2 multicolored r-uniform 2-factors in this decomposition where lr is the number of elements of order r in G, 3 ≤ r ≤ n. This provides a generalization of a recent result due to Constantine which... 

Let R be a left artinian central F-algebra, T(R) = J(R) + [R,R], and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of R = R/J(R) is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson. © Inst. Math. CAS 2001  

In this paper we investigate two conjectures proposed in (Graphs Combin. 13 (1997) 305-314). The first one is uniquely totally colorable (UTC) conjecture which states: Empty graphs, paths, and cycles of order 3k, k a natural number, are the only UTC graphs. We show that if G is a UTC graph of order n, then Δn/2+1, where Δ is the maximum degree of G. Also there is another question about UTC graphs that appeared in (Graphs Combin. 13 (1997) 305-314) as follows: If a graph G is UTC, is it true that in the proper total coloring of G, each color is used for at least one vertex? We prove that if G is a UTC graph of order n and in the proper total coloring of G, there exists a color which did not... 

Polystyrene/organoclay nanocomposites were prepared by melt intercalation method in this research. Morphology, tensile and impact properties and deformation mechanism of the samples were studied. To study the structure of nanocomposites, X-ray diffraction and transmission electron microscopy techniques are utilized. The deformation mechanisms of different samples were examined via reflected and transmitted optical microscopy. The results reveal that incorporation of organoclay affects structure, mechanical properties and deformation mechanism of nanocomposite. Introduction of organoclay can facilitate initiation and growth of crazing mechanism in polystyrene at both conditions of loadings,... 

Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x, y∈R are adjacent if and only if x+y∈Z(R). Let the regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a commutative ring are contained in the set {3, 4, ∞}. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). Also, we prove that if R is a reduced left Noetherian ring and 2∈Z(R), then the chromatic number and the clique number of Reg(Γ(R)) are the... 

Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by (TΓ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y in R are adjacent if and only if x + y Z(R). Let the regular graph of R, Reg (Γ(R)), be the induced subgraph of T(Γ (R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set { 3, 4,} . In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if R is a reduced left Noetherian ring and 2 Z(R), then the chromatic number and the clique number of Reg... 

Let be a ring and be the set of all non-trivial left ideals of. The intersection graph of ideals of, denoted by, is a graph with the vertex set and two distinct vertices and are adjacent if and only if. In this paper, we classify all rings (not necessarily commutative) whose domination number of the intersection graph of ideals is at least 2. Moreover, some results on the intersection graphs of ideals of matrix algebras over a finite field are given. For instance, we determine the domination number, the clique number and the independence number of. We prove that if is a positive integer and, then the domination number of is. Among other results, we show that if, where is a positive integer... 

For every natural number h, a graph G is said to be h-magic if there exists a labelling l:E(G)→Zh{0} such that the induced vertex set labelling l+:V(G)→Zh defined byl+(v)=∑uv∈E(G)l(uv), is a constant map. When this constant is zero, it is said that G admits a zero-sum h-magic labelling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h∈N such that G admits an h-zero-sum magic labelling. In 2007, E. Salehi determined the null set of complete bipartite graphs. In this paper we generalize this result by obtaining the null set of complete multipartite graphs  

Let S⊆C*=C{0} and A∈Mn(C). The matrix A is called an S-GHMn if A∈Mn(S) and AA*=Diag(λ1,... λn), for some positive numbers λi, i=1,... n. In this paper we provide some necessary conditions on n for the existence of an S-GHMn over a finite set S. We conjecture that for every positive integer n, there exists a {±1, ±2, ±3}-GHMn  

Let R be a commutative ring with unity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2), ℤ2[x, y]/(x, y)2, ℤ4[x]/(2x, x 2). We prove that for every... 

For a set S of positive integers, a spanning subgraph F of a graph G is called an S-factor of G if degF(x) ∈ S for all vertices x of G, where degF(x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an even integer such that 2 ≤ k < r/2 or an odd integer such that r/3 ≤ k < r/2. Then every r-regular graph G has a {k, r-k}-factor. Moreover, for every edge e of G, G has a {k, r-k}-factor containing e and another {k, r-k}-factor avoiding e