Boolean ring

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In mathematics, a Boolean ring R is a ring for which x² = x for all x in R; that is, R consists only of idempotent elements. These rings arise from (and give rise to) Boolean algebras.

Contents 1 Examples 2 Relation to Boolean algebras 3 Facts 4 References

[edit] Examples

One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite subsets of X, again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

[edit] Relation to Boolean algebras

Given a Boolean ring R, for x and y in R we can define

x ∧ y = xy,

x ∨ y = x + y − xy,

~x = 1 + x.

These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra (for consistency, we use x + y − xy, though, as noted under Facts below, one could write x + y + xy because it follows from the definition above that the identity x = −x holds in these rings). Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:

xy = x ∧ y,

x + y = (x ∨ y) ∧ ~(x ∧ y).

A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.

[edit] Facts

Every Boolean ring R satisfies x + x = 0 for all x in R, because we know

x + x = (x + x)² = x² + 2x² + x² = x + 2x + x = x + x + x + x

and since <R,+> is an abelian group, we can subtract x + x from both sides of this equation, which gives x + x = 0. A similar proof shows that every Boolean ring is commutative:

x + y = (x + y)² = x² + xy + yx + y² = x + xy + yx + y

and this yields xy + yx = 0, which means xy = −yx = yx (using the first property above).

The property x + x = 0 shows that any Boolean ring is an associative algebra over the field F₂ with two elements, in just one way. In particular, any finite Boolean ring has as cardinality a power of two. Not every associative algebra with one over F₂ is a Boolean ring: consider for instance the polynomial ring F₂[X].

The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and at the same time a Boolean ring, so it must be isomorphic to the field F₂, which shows the maximality of P. Since maximal ideals are always prime, we conclude that prime ideals and maximal ideals coincide in Boolean rings.

[edit] References

• Atiyah, Michael Francis & Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8it:Anello booleano

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