View source for Gelfand ring

{{Short description|Algebraic structure}}
In [[mathematics]], a '''Gelfand ring''' is a [[ring (mathematics)|ring]] ''R'' with identity such that if ''I'' and ''J'' are distinct right [[ideal (ring theory)|ideal]]s 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''.   {{harvtxt|Mulvey|1979}} introduced them as rings for which one could [[mathematical proof|prove]] a generalization of [[Gelfand duality]], and named them after [[Israel Gelfand]].<ref>{{citation|last=Mulvey|first=Christopher J.|title=A generalisation of Gelʹfand duality.|journal=J. Algebra|volume=56|issue=2|pages=499–505|year=1979|doi=10.1016/0021-8693(79)90352-1|mr=0528590|doi-access=free}}</ref>
In the [[commutative ring|commutative]] case, Gelfand rings can also be characterized as the rings such that, for every {{mvar|a}} and {{mvar|b}} summing to {{math|1}}, there exists {{mvar|r}} and {{mvar|s}} such that
:<math>(1+ra)(1+sb) = 0</math>.
Moreover, their [[prime spectrum]] [[deformation retract]]s onto the [[maximal spectrum]].<ref>{{Cite journal|last=Contessa|first=Maria|date=1982-01-01|title=On pm-rings|url=https://doi.org/10.1080/00927878208822703|journal=Communications in Algebra|volume=10|issue=1|pages=93–108|doi=10.1080/00927878208822703|issn=0092-7872}}</ref><ref>{{Cite web|title=algebraic geometry - When does the prime spectrum deformation retract into the maximal spectrum?|url=https://math.stackexchange.com/q/1586745 |access-date=2020-10-16|website=Mathematics Stack Exchange}}</ref>
==References==
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[[Category:Ring theory]]
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