We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Extending Abelian Rings: A Generalized Approach.
- Authors
Saad, Muhammad; Zailaee, Majed
- 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]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.
- Subjects
MATRIX rings; SEMIRINGS (Mathematics); IDEMPOTENTS; COMMUTATION (Electricity)
- Publication
European Journal of Pure & Applied Mathematics, 2024, Vol 17, Issue 2, p736
- ISSN
1307-5543
- Publication type
Article
- DOI
10.29020/nybg.ejpam.v17i2.5066