Rings of quotients of incidence algebras and path algebras

Abstract

We compute the maximal right/left/symmetric rings of quotients of finite-dimensional incidence and graph algebras. We show that these rings of quotients are Morita equivalent to incidence algebras and path algebras, respectively, with respect to simpler, well determined partially ordered sets and finite quivers, respectively. IntroductionBreaking away from the classical rings of quotients a more general kind of ring of quotients was introduced by Utumi [18] leading him to the notion of the maximal ring of quotients. This concept had been studied by Johnson [8] in the case of a nonsingular ring. For a noncommutative ring R, there are a maximal right ring of quotients Q r max (R) and a maximal left ring of quotients Q l max (R). The conditions under which these two rings of quotients coincide were determined by Utumi in [19] using as a tool a maximal symmetric ring of quotients. However Utumi did not undertake any systematic study of this ring of quotients. The maximal symmetric ring of quotients appeared later in a work of Schelter [13], where it is considered in the framework of torsion theories on rings. The first systematic study of the maximal symmetric ring of quotients

Algebras of quotients of path algebras

“…Apply [22,Proposition 3.4] to the opposite graph of E (the one obtained from E by changing ranges to sources and sources to ranges) and [2, Proposition 3.5] to show that Q r max (KE) = L K (E). The fact of L K (E) be semiprime and artinian and an algebra of right quotients of A (because it is an algebra of right quotients of KE, by Proposition 2.2) implies, by [14,Corollary 3.4], that it is the classical algebra of right quotients of A.…”

Rings of quotients of some infinite dimensional path algebras

“…For a field F and a quiver T, denote by FT its associated path algebra (see [5] and [9] for more details). DEFINITIONS 1.4.…”

“…This ring construction was first used in Utumi's work [14] (although he did not define it formally), later it was studied by Schelter [11], where it appears as a particular case of a more general concept of two-sided localization of R with respect to Gabriel filters of right and left ideals of R. The ring Q a (R) has been used in several papers, see for instance [2] and [4] and it is defined as More recently, Lanning [8] has performed a deep study of the maximal symmetric ring of quotients. In a previous paper [9] we have studied some ring theoretical properties of the maximal symmetric ring of quotients, computed explicitly the maximal symmetric ring of quotients for certain finite dimensional algebras and in [10] we have established a categorical framework for two-sided localizations. The benefit of the two-sided rings of quotients rests on the facts that they preserve the symmetries of the ring and that they are useful in the study of two-sided concepts related to the ring.…”

“…In [9] we computed the maximal symmetric ring of quotients of a path algebra when the quiver T was finite (i.e. |To| < oo and |Ti| < oo) and had no oriented cycles.…”

The maximal C*-algebra of quotients as an operator bimodule

“…The maximal C*-algebra of quotients, Q max (A), of a (unital) C*-algebra A was introduced in [3] as a C*-analytic analogue of the maximal symmetric ring of quotients of a non-singular ring studied, e.g., in [9] and [10]. As a C*-algebra of quotients it shares some of the properties of the local multiplier algebra M loc (A) of A [2]; for instance, it arises as the completion of the bounded part of its algebraic counterpart and it can be canonically embedded into the injective envelope I(A) of A.…”