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A ring [Formula: see text] is called semiboolean if [Formula: see text] is boolean and idempotents lift modulo [Formula: see text], where [Formula: see text] denotes the Jacobson radical of [Formula: see text]. In this paper, we define [Formula: see text]-boolean rings as a generalization of semiboolean rings. A ring [Formula: see text] is said to be J-boolean if [Formula: see text] is boolean. Various basic properties of these rings are obtained. The [Formula: see text]-boolean group rings and skew group rings have been studied. It is investigated whether the results obtained for [Formula: see text]-boolean group rings also hold for the skew group rings.

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