Let I(X, R) be an incidence algebra where X is a locally finite partially ordered set and R is a commutative ring with identity. Our intention is to construct the maximal or Utumi ring of quotients, which is defined for any ring T, of an incidence algebra I(X, R). Following the alternate description given by Findlay-Lambek, we use the dense ideals of the ring to construct its maximal ring of quotients. Since, in general, it is hard to determine all the dense ideals of a ring, instead we construct a basis of dense ideals that form the Gabriel topology on the dense ideals. Our starting point will be the fact that any dense ideal is also an essential ideal. In this research, results about essential ideals are used to obtain description of the dense ideals of an incidence algebra. Also, the necessary and sufficient conditions for I(X, R) to have a minimal dense ideal are stated. Further, this result is used to compute the maximal quotient ring of some incidence algebras.
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机译：假设 I （ X，R ）是一个发生代数，其中 X 是局部有限的部分有序集，而 R 是具有身份的交换环。我们的目的是构造最大代数或Utumi商环，该代数定义为发生代数 I （ X，R ）。根据Findlay-Lambek给出的替代描述，我们使用环的密集理想构造其商的最大环。由于通常很难确定环的所有稠密理想，因此我们构建了稠密理想的基础，该稠密理想形成了在稠密理想上的Gabriel拓扑。我们的出发点将是以下事实：任何密集的理想也是必不可少的理想。在这项研究中，关于基本理想的结果用于获得入射代数的密集理想的描述。还说明了 I （ X，R ）具有最小密集理想的必要和充分条件。此外，该结果用于计算某些入射代数的最大商环。
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