GORENSTEIN TILED ORDERS

Abstract

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.