Boolean ring

A Boolean ringMathworldPlanetmath is a ring R that has a multiplicative identityPlanetmathPlanetmath, and in which every element is idempotentPlanetmathPlanetmath, that is,

x2=x for all xR.

Boolean rings are necessarily commutativePlanetmathPlanetmathPlanetmathPlanetmath ( Also, if R is a Boolean ring, then x=-x for each xR.

Boolean rings are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to Boolean algebrasMathworldPlanetmath (or Boolean lattices ( Given a Boolean ring R, define xy=xy and xy=x+y+xy and x=x+1 for all x,yR, then (R,,,,0,1) is a Boolean algebra. Given a Boolean algebra (L,,,,0,1), define xy=xy and x+y=(xy)(xy), then (L,,+) is a Boolean ring. In particular, the category of Boolean rings is isomorphicPlanetmathPlanetmathPlanetmath to the category of Boolean lattices.


As mentioned above, every Boolean algebra can be considered as a Boolean ring. In particular, if X is any set, then the power setMathworldPlanetmath 𝒫(X) forms a Boolean ring, with intersectionMathworldPlanetmath as multiplication and symmetric differenceMathworldPlanetmathPlanetmath as addition.

Let R be the ring 2×2 with the operationsMathworldPlanetmath being coordinate-wise. Then we can check:

(1,1)×(1,1) = (1,1)
(1,0)×(1,0) = (1,0)
(0,1)×(0,1) = (0,1)
(0,0)×(0,0) = (0,0)

the four elements that form the ring are idempotent. So R is Boolean.

Title Boolean ring
Canonical name BooleanRing
Date of creation 2013-03-22 12:27:28
Last modified on 2013-03-22 12:27:28
Owner yark (2760)
Last modified by yark (2760)
Numerical id 24
Author yark (2760)
Entry type Definition
Classification msc 06E99
Classification msc 03G05
Related topic Idempotency
Related topic BooleanLattice
Related topic BooleanIdeal