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Extending Abelian Rings: A Generalized Approach.
- Source :
-
European Journal of Pure & Applied Mathematics . Apr2024, Vol. 17 Issue 2, p736-752. 17p. - Publication Year :
- 2024
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATRIX rings
*SEMIRINGS (Mathematics)
*IDEMPOTENTS
*COMMUTATION (Electricity)
Subjects
Details
- Language :
- English
- ISSN :
- 13075543
- Volume :
- 17
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- European Journal of Pure & Applied Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 177977458
- Full Text :
- https://doi.org/10.29020/nybg.ejpam.v17i2.5066