Skew incidence rings and the isomorphism problem

TL;DR

This paper studies properties of skew incidence rings constructed from posets and rings with endomorphisms, and establishes conditions under which isomorphic skew incidence rings imply isomorphic underlying posets.

Contribution

It characterizes algebraic properties of skew incidence rings and proves that isomorphisms between such rings reflect isomorphisms of the underlying posets under certain conditions.

Findings

Description of invertible elements, idempotents, Jacobson radical, and center of skew incidence rings.

Proof that isomorphic skew incidence rings imply isomorphic posets when only trivial idempotents exist.

Conditions under which the isomorphism problem for skew incidence rings reduces to poset isomorphism.

Abstract

Let $X$ be a finite partially ordered set, $R$ an associative unital ring and $\sigma$ an endomorphism of $R$. We describe some properties of the skew incidence ring $I(X,R,\sigma)$ such as invertible elements, idempotents, the Jacobson radical and the center. Moreover, if the skew incidence rings $I(X,R,\sigma)$ and $I(Y,S,\tau)$ are isomorphic and the only idempotents of $R,S$ are the trivial ones, we show that the partially ordered sets $X$ and $Y$ are isomorphic.