The Singular Ideal and the Socle of Incidence Algebras

TL;DR

This paper investigates the structure of incidence algebras over rings, specifically computing their socle and singular ideal in relation to the base ring's properties, for certain partially ordered sets.

Contribution

It provides explicit formulas for the socle and singular ideal of incidence algebras based on the socle and singular ideal of the underlying ring, for specific classes of posets.

Findings

Explicit computation of the socle of incidence algebras.

Explicit computation of the singular ideal of incidence algebras.

Results depend on properties of the base ring and the poset structure.

Abstract

Let $R$ be a ring with identity and $I(X,R)$ be the incidence algebra of a locally finite partially ordered set $X$ over $R.$ In this paper, we compute the socle and the singular ideal of the incidence ring for some $X$ in terms of the socle of $R$ and the singular ideal of $R$, respectively.