Gelfand ring explained

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. 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

( 1+ ra)( 1+ sb) = 0 . Moreover, their prime spectrum deformation retracts onto the maximal spectrum.^{[1] [2]}

Notes and References

1. Contessa. Maria. 1982-01-01. On pm-rings. Communications in Algebra. 10. 1. 93–108. 10.1080/00927878208822703. 0092-7872.
2. Web site: algebraic geometry - When does the prime spectrum deformation retract into the maximal spectrum?. 2020-10-16. Mathematics Stack Exchange.