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 a and b summing to 1, there exists r and s such that . Moreover, their prime spectrum deformation retracts onto the .