JP Journal of Algebra, Number Theory and Applications
Volume 39, Issue 3, Pages 261 - 276
(June 2017) http://dx.doi.org/10.17654/NT039030261 |
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WEAKLY SEMI-BOOLEAN UNITAL RINGS
Peter V. Danchev
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Abstract: With regard to Problem 5 from [10], we completely describe weakly semi-Boolean rings. We prove the following results: (1) a ring Ris weakly semi-Boolean if, and only if, idempotents lift modulo the Jacobson radical and is isomorphic to either B, or or where Bis a Boolean ring. This criterion is then applied to find a necessary and sufficient condition when the commutative group ring is weakly semi-Boolean; (2) for any ring R, the matrix ring is never weakly semi-Boolean, provided (3) for any commutative ring R, the power series ring is weakly semi-Boolean if, and only if, Ris weakly semi-Boolean.
These achievements extend results due to Nicholson-Zhou in [20] and Danchev-McGovern in [10]. |
Keywords and phrases: clean rings, semi-Boolean rings, weakly semi-Boolean rings, idempotents, Jacobson radical. |
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