Summary: A Generalization of Boolean Rings
Abstract: A Boolean ring satisfies the identity x2 = x which, of course, implies the identity
x2y - xy2 = 0. With this as motivation, we define a subBoolean ring to be a ring R which satisfies
the condition that x2y-xy2 is nilpotent for certain elements x, y of R. We consider some conditions
which imply that the subBoolean ring R is commutative or has a nil commutator ideal.
Throughout, R is a ring, not necessarily with identity, N the set of nilpotents, C the
center, and J the Jacobson radical of R. As usual, [x, y] will denote the commutator xy-yx.
Definition. A ring R is called subBoolean if
y - xy2
N for all x, y in R \ (N J C).
The class of subBoolean rings is quite large, and contains all Boolean rings, all commutative
rings, all nil rings, and all rings in which J = R. On the other hand, a subBoolean ring need
not be Boolean or even commutative. Indeed, the ring