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Gelfand ring

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Algebraic structure

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.

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 + r a ) ( 1 + s b ) = 0 {\displaystyle (1+ra)(1+sb)=0} .

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

References

  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.
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