Exponent matrices and Frobenius rings

We give a survey of results connecting the exponent matrices with Frobenius rings. In particular, we prove that for any parmutation σ ∈ Sn there exists a countable set of indecomposable Frobenius semidistributive rings Am with Nakayama permutation σ

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

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

We prove that the quiver of tiled order over a discrete valuation ring is strongly connected and simply laced. With such quiver we associate a finite ergodic Markov chain. We introduce the notion of the index in A of a right noetherian semiperfect ring A as the maximal real eigen-value of its adjacency matrix. A tiled order Λ is integral if in Λ is an integer. Every cyclic Gorenstein tiled order is integral. In particular, in Λ = 1 if and only if Λ is hereditary. We give an example of a non-integral Gorenstein tiled order. We prove that a reduced (0, 1)-order is Gorenstein if and only if either inΛ = w(Λ) = 1, or inΛ = w(Λ) = 2, where w(Λ) is a width of Λ