Mathematical Notes, vol. 72, no. 3, 2002, pp. 428–432. Translated from Matematicheskie Zametki, vol. 72, no. 3, 2002, pp. 468–471. Original Russian Text Copyright c 2002 by M. A. Dokuchaev, V. V. Kirichenko, Zh. T. Chernousova. BRIEF COMMUNICATIONS Tiled Orders and Frobenius Rings M. A. Dokuchaev, V. V. Kirichenko, and Zh. T. Chernousova Received February 26, 2002 Key words: tiled order, Frobenius ring, discretely normed ring, Nakayama permutation. In the present paper we construct a countable set of Frobenius quotient rings with identity Nakayama permutation for any reduced tiled order over a given discrete valuation ring. In partic- ular, for any finite poset P =(p 1 ,...,p n ), we construct countably many Frobenius rings F m (P ) such that the quivers Q(F m (P )) of the rings F m (P ) coincide. Denote by P max the set of all maximal elements of P , by P min the set of all minimal elements of P , and by P max × P min their Cartesian product. To state the relationship between the quiver Q(F m (P )) of a ring and the poset P , we recall the definition of the diagram of a poset P [1, p. 233]. The diagram of a poset P =(p 1 ,...,p n ) is the quiver Q(P ) with the set of vertices VQ(P )= {1 ,...,n} and the set of arrows AQ(P ), where an arrow from a vertex i to a vertex j exists if and only if p i <p j , and, if p i ≤ p k ≤ p j , then either k = i or k = j . The quiver Q(F m (P )) is obtained from the diagram Q(P ) by adding the arrows σ ij for any (p i ,p j ) ∈ P max × P min . Therefore, if P is a totally ordered set of n elements, then Q(F m (P )) is the simple cycle C n , and hence all rings F m (P ) are serial. 1. TILED ORDERS All rings considered in the paper are assumed to be associative and to have a nonzero identity element. For all necessary information about Frobenius and quasi-Frobenius rings, see [2, Chap. 6]. The definition of the Nakayama permutation can also be found in [2, p. 426]. Denote by M n (B) the ring of all square matrices of order n with the entries in a ring B . Let E =(α ij ) ∈ M n (Z), where Z is the ring of integers. Definition 1.1. An integral matrix E =(α ij ) is said to be a matrix of exponents if α ij +α jk ≥ α ik and α ii = 0 for i,j,k =1 ,...,n . A matrix of exponents E =(α ij ) is said to be reduced if α ij + α ji > 0. Let O be a discrete valuation ring with a unique maximal ideal M = πO = Oπ and with the classical skew field of quotients D . From this ring and a matrix of exponents E =(α ij ) we construct an order Λ in M n (D) of the following form: Λ=    O π α 12 O ... π α 1n O π α 21 O O ... π α 2n O ........................... π α n1 O π α n2 O ... O    . (1) An order of the form (1) is said to be a tiled order over the discrete valuation ring O . Since no other orders are considered in the paper, we refer to the orders of the form (1) as tiled orders, or simply orders, and write Λ = {O , E = E (Λ)} . 428 0001-4346/2002/7234-0428$27.00 c 2002 Plenum Publishing Corporation