Jacobson Radical in Artinian Z-Algebras-Nilpotency and Centrality
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| Natura: | Recurso digital |
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Zenodo
2025
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| _version_ | 1866902028262834176 |
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| author | Chuang, William Huanshan |
| author_facet | Chuang, William Huanshan |
| contents | <p>These notes prove that the Jacobson radical J(A) of an Artinian ring (e.g., a finite ring or a finite-dimensional Z-algebra) is nilpotent. After reviewing Z-algebras, Artinian chain conditions, nilpotent elements/ideals, the center Z(A), and the Jacobson radical, the main theorem is shown via stabilization of the powers J(A)^n and the semisimplicity of A/J(A). By Wedderburn–Artin, A/J(A) is a finite direct sum of matrix algebras over division rings, which contains no nonzero nilpotent ideals; hence J(A)^N = 0 for some N. In commutative settings the image of Z lies in the center automatically; in noncommutative rings the radical need not be central, but its nilpotency drives the structural “peel-off” A → A/J(A). Examples and applications to structure theory are included.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17352562 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Jacobson Radical in Artinian Z-Algebras-Nilpotency and Centrality Chuang, William Huanshan <p>These notes prove that the Jacobson radical J(A) of an Artinian ring (e.g., a finite ring or a finite-dimensional Z-algebra) is nilpotent. After reviewing Z-algebras, Artinian chain conditions, nilpotent elements/ideals, the center Z(A), and the Jacobson radical, the main theorem is shown via stabilization of the powers J(A)^n and the semisimplicity of A/J(A). By Wedderburn–Artin, A/J(A) is a finite direct sum of matrix algebras over division rings, which contains no nonzero nilpotent ideals; hence J(A)^N = 0 for some N. In commutative settings the image of Z lies in the center automatically; in noncommutative rings the radical need not be central, but its nilpotency drives the structural “peel-off” A → A/J(A). Examples and applications to structure theory are included.</p> |
| title | Jacobson Radical in Artinian Z-Algebras-Nilpotency and Centrality |
| url | https://doi.org/10.5281/zenodo.17352562 |