Extending Abelian Rings: A Generalized Approach

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]ⁿe=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 Tₙ(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.