Gelfand ring

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&hairsp;R&hairsp;j = 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
 * $$(1+ra)(1+sb) = 0$$.

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