Maximal ring of quotients of an incidence algebra

Weak incidence algebra and maximal ring of quotients

“…The main aim of Section 4 is to prove some results that can help in studying the maximal ring of quotients of an I * (X, R). Similar work has been done in a recent paper [2] for certain classes of incidence algebras. In [7], Spiegel determines some essential ideals of an incidence algebra of a locally finite, partially f ∈ S, support of f , denoted by suppt(f ), equals {(x, y) : f (x,y) ≠ 0}, the cardinality of suppt(f ) is called the weight of f and we denote it by wt(f ).…”

“…is a right ideal of S. In [2], maximal rings of quotients of certain incidence algebras have been discussed. Here we intend to prove some results that can help in studying the maximal rings of quotients of S. Spiegel [7] has determined certain classes of essential ideals of an incidence algebra of a locally finite, preordered set.…”

“…We now discuss matrix representations of Hom S (T [ [2]. Let X be any partially ordered set such that for any x ∈ X, there exists a maximal element z ≥ x and L z = {y ∈ X : y ≤ z} is finite.…”

Rings of quotients of incidence algebras and path algebras

“…The right and left maximal rings of quotients of finite-dimensional incidence algebras are well-known [1,9], so we will focus our attention on the maximal symmetric ring of quotients of incidence algebras of finite ordered sets X and path algebras of finite quivers Γ without oriented cycles. The case of locally finite quivers will be considered elsewhere.…”

“…Now we summarize the contents of this paper. In Section 1 we review various known facts on rings of quotients and we observe that for a full idempotent e in any unital ring R we have Q σ (eRe) ∼ = eQ σ (R)e. Then we give basic definitions on incidence algebras I (X, F ) of finite partially ordered sets X, recall the computation of their maximal right rings of quotients [1,9] and present a first connection between incidence algebras and path algebras [5]. Given a finite partially ordered set Γ 0 there is a natural finite quiver Γ without oriented cycles and a surjective algebra homomorphism Φ : F Γ → I (Γ 0 , F ) which is an isomorphism if and only if I (Γ 0 , F ) is hereditary which in turn is equivalent to Γ 0 being a tree in the sense of partially ordered sets.…”