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 R j = 0, i is not in I, and j is not in J. Mulvey (1979) introduced them as rings for which one could prove a generalization of Gelfand duality, and named them after Israel Gelfand.^[1]

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.^[2]^[3]

References

^ Mulvey, Christopher J. (1979), "A generalisation of Gelʹfand duality.", J. Algebra, 56 (2): 499–505, doi:10.1016/0021-8693(79)90352-1, MR 0528590
^ Contessa, Maria (1982-01-01). "On pm-rings". Communications in Algebra. 10 (1): 93–108. doi:10.1080/00927878208822703. ISSN 0092-7872.
^ "algebraic geometry - When does the prime spectrum deformation retract into the maximal spectrum?". Mathematics Stack Exchange. Retrieved 2020-10-16.

This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.

[1] Mulvey, Christopher J. (1979), "A generalisation of Gelʹfand duality.", J. Algebra, 56 (2): 499–505, doi:10.1016/0021-8693(79)90352-1, MR 0528590

[2] Contessa, Maria (1982-01-01). "On pm-rings". Communications in Algebra. 10 (1): 93–108. doi:10.1080/00927878208822703. ISSN 0092-7872.

[3] "algebraic geometry - When does the prime spectrum deformation retract into the maximal spectrum?". Mathematics Stack Exchange. Retrieved 2020-10-16.

[1]

[2]

[3]