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].

clean rings, semi-Boolean rings, weakly semi-Boolean rings, idempotents, Jacobson radical.