Algebra and Discrete Mathematics, 2012, Vol. 14, № 2

Andrei Vladimirovich Roiter. To the 75th anniversary

2012-01-01T00:00:00Z

Andrei Vladimirovich Roiter. To the 75th anniversary Drozd, Yu.; Kirichenko, V.; Krugliak, S.; Kleiner, M.; Bondarenko, V.; Ovsienko, S.

Projective resolution of irreducible modules over tiled order

2012-01-01T00:00:00Z

Projective resolution of irreducible modules over tiled order Zhuravlev, V.; Zhuravlyov, D. We indicate the method for computing the kernels of projective resolution of irreducible module over tiled order. On the base of this method we construct projective resolution of irreducible module and calculate the global dimension of tiled order. The evident view of kernels of projective resolution allows to check easily the regularity of tiled order.

The detour hull number of a graph

2012-01-01T00:00:00Z

The detour hull number of a graph Santhakumaran, A.P.; Ullas Chandran, S.V. For vertices u and v in a connected graph G = (V, E), the set ID[u, v] consists of all those vertices lying on a u−v longest path in G. Given a set S of vertices of G, the union of all sets ID[u, v] for u, v ∈ S, is denoted by ID[S]. A set S is a detour convex set if ID[S] = S. The detour convex hull [S]D of S in G is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among the subsets S of V with [S]D = V. A set S of vertices is called a detour set if ID[S] = V. The minimum cardinality of a detour set is the detour number dn(G) of G. A vertex x in G is a detour extreme vertex if it is an initial or terminal vertex of any detour containing x. Certain general properties of these concepts are studied. It is shown that for each pair of positive integers r and s, there is a connected graph G with r detour extreme vertices, each of degree s. Also, it is proved that every two integers a and b with 2 ≤ a ≤ b are realizable as the detour hull number and the detour number respectively, of some graph. For each triple D, k and n of positive integers with 2 ≤ k ≤ n − D + 1 and D ≥ 2, there is a connected graph of order n, detour diameter D and detour hull number k. Bounds for the detour hull number of a graph are obtained. It is proved that dn(G) = dh(G) for a connected graph G with detour diameter at most 4. Also, it is proved that for positive integers a, b and k ≥ 2 with a < b ≤ 2a, there exists a connected graph G with detour radius a, detour diameter b and detour hull number k. Graphs G for which dh(G) = n − 1 or dh(G) = n − 2 are characterized.

On radical square zero rings

2012-01-01T00:00:00Z

On radical square zero rings Ringel, C.M.; Xiong, B.-L. Let Λ be a connected left artinian ring with radical square zero and with n simple modules. If Λ is not self-injective, then we show that any module M with Exti(M, Λ) = 0 for 1 ≤ i ≤ n + 1 is projective. We also determine the structure of the artin algebras with radical square zero and n simple modules which have a non-projective module M such that Exti(M, Λ) = 0 for 1 ≤ i ≤ n.