Project On A Generalization Of Boolean Rings

In this paper we introduce the concept of Boolean ring and 𝑅𝑅 satisfy the identity 𝑥𝑥2=𝑥𝑥 which, of course, implies the identity 𝑥𝑥2𝑦𝑦−𝑥𝑥𝑦𝑦2=0. With this as motivation, we define a Boolean like ring and subBoolean ring 𝑅𝑅 to be a ring 𝑅𝑅 which satisfies the condition that 𝑥𝑥2𝑦𝑦−𝑥𝑥𝑦𝑦2 is nilpotent for certain elements 𝑥𝑥,𝑦𝑦 in 𝑅𝑅. A strongly Boolean ring is a ring which 𝑥𝑥2𝑦𝑦=𝑥𝑥𝑦𝑦2 for some elements 𝑥𝑥,𝑦𝑦 in 𝑅𝑅. The commutativity behavior of such rings is considered. Also, certain conditions which imply that these rings have a nil commutator ideal are established. We consider some conditions which imply that the subBoolean ring 𝑅𝑅 is commutative or has a nil commutator ideal. We also prove that a generalized Boolean ring with central idempotents must be nil or commutative. We further consider conditions which imply the commutativity of a generalized Boolean ring

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