Extending Abelian Rings: A Generalized Approach

Abstract

We introduce a novel framework for assessing the centrality of idempotents within a ring by presenting a general concept that assigns a degree of centrality. This approach aligns with the previously established notions of semicentral and q-central idempotents by Birkenmeier and Lam. Specifically, we define an idempotent $e$ in a ring $R$ to be $n$-central, where $n$ is a positive integer, if $[e,R{]}^{n}e=0$, where $[x,y]$ represents the additive commutator $xy-yx$. If every idempotent in a ring $R$ is $n$-central, we refer to $R$ as $n$-Abelian. Our study lays the groundwork by presenting foundational results that support this concept and examines key features of $n$-central idempotents essential for appropriately categorizing $n$-Abelian rings among various generalizations of Abelian rings introduced in prior literature. We provide examples of $n$-central idempotents that do not fall under the categories of semicentral or $q$-central. Furthermore, we demonstrate that the ring of upper matrices ${\mathbb{T}}_n(R)$, where $R$ is Abelian, is an $n$-abelian. We also prove that a ring where all of its idempotents are $n$-central is an exchange ring if and only if the ring is clean.