Gelfand ring

Template:Short description In mathematics, a Gelfand ring is a ring R with identity such that if I and J are distinct right ideals then there are elements i and j such that i R j = 0, i is not in I, and j is not in J. Template:Harvtxt 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 Template:Mvar and Template:Mvar summing to Template:Math, there exists Template:Mvar and Template:Mvar such that

$(1+ra)(1+sb)=0$.

Moreover, their prime spectrum deformation retracts onto the maximal spectrum.