On semiperfect rings of injective dimension one

We give a characterization of right Noetherian semiprimesemiperfect and semidistributive rings with inj. dimAAA 6 1

Gorenstein Quivers

We introduce a notion of Gorenstein quiver associated witha Gorenstein matrix. We study properties of such quivers. In partic-ular, we show that any such quiver is strongly connected and simplylaced. We use Perron-Frobenius theory of non-negative matrices forcharacterization of isomorphic Gorenstein quivers

Tiled orders over discrete valuation rings, finite Markov chains and partially ordered sets. II

The main concept of this part of the paper is that of a reduced exponent matrix and its quiver, which is strongly connected and simply laced. We give the description of quivers of reduced Gorenstein exponent matrices whose number s of vertices is at most 7. For 2 ≤ 6 s ≤ 5 we have that all adjacency matrices of such quivers are multiples of doubly stochastic matrices. We prove that for any permutation σ on n letters without fixed elements there exists a reduced Gorenstein tiled order Λ with σ(ε) = σ. We show that for any positive integer k there exists a Gorenstein tiled order Λk with inΛk = k. The adjacency matrix of any cyclic Gorenstein order Λ is a linear combination of powers of a permutation matrix Pσ with non-negative coefficients, where σ = σ(Λ). If A is a noetherian prime semiperfect semidistributive ring of a finite global dimension, then Q(A) be a strongly connected simply laced quiver which has no loops

Gorenstein matrices

Let A = (aij ) be an integral matrix. We say that A is (0, 1, 2)-matrix if aij ∈ {0, 1, 2}. There exists the Gorenstein (0, 1, 2)-matrix for any permutation σ on the set {1, . . . , n} without fixed elements. For every positive integer n there exists the Gorenstein cyclic (0, 1, 2)-matrix An such that inx An = 2. If a Latin square Ln with a first row and first column (0, 1, . . . n − 1) is an exponent matrix, then n = 2m and Ln is the Cayley table of a direct product of m copies of the cyclic group of order 2. Conversely, the Cayley table Em of the elementary abelian group Gm = (2)×. . .×(2) of order 2 m is a Latin square and a Gorenstein symmetric matrix with first row (0, 1, . . . , 2 m − 1) and σ(Em) = 1 2 3 . . . 2 m − 1 2m 2 m 2 m − 1 2m − 2 . . . 2 1

Quivers of 3×3 exponent matrices

We show how to use generating exponent matrices to study the quivers of exponent matrices. We also describe the admissible quivers of 3×3 exponent matrices

Quivers of 3×3-exponent matrices

We show how to use generating exponent matrices to study the quivers of exponent matrices. We also describe the admissible quivers of 3×3 exponent matrices

Projective resolution of irreducible modules over tiled order

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

Tiled orders of width 3

We consider projective cover over tiled order and calculate the kernel of epimorphism from direct sum of submodules of distributive module to their sum