Gelfand ring

In mathematics, a Gelfand ring is an associative ring "R" with identity such that if "I" and "J" are distinct right ideals then there are elements "i" and "j" such that "iRj"=0, "i" is not in "I", and "j" is not in "J". introduced them as rings for which one could prove a generalization of Gelfand duality, and named them after Israel Gelfand. 
In the commutative case, Gelfand rings can also be characterized as the rings such that, for every and summing to , there exists and such that
Moreover, their prime spectrum deformation retracts onto the maximal spectrum. 