Boolean ring

From Academic Kids

In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. These rings arise from (and give rise to) Boolean algebras. One example is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection.

Relation to Boolean algebras

If we define

xy = xy,
xy = x + yxy,
~x = 1 + x

then these 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 = xy,
x + y = (xy) ∧ ~(xy).

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.

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

and we can subtract x + x from both sides of this equation. 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.it:Anello booleano

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