Boolean Algebras and Stone Duality
Boolean Rings and Idempotent Rings
Rings in which every element is idempotent, their forced properties, and their status as an equationally defined class.
Learning objectives
- Define Boolean ring and derive its basic properties
- Prove commutativity and characteristic 2 from idempotence
- Situate Boolean rings as a variety
Definition and first consequences
A ring with unit in which every element is idempotent: x2 ≈ x.
In a Boolean ring, x + x = 0 for every x; equivalently x = −x.
Expand (x + x)2 = x2 + x2 + x2 + x2 = 4x. Idempotence of x + x gives 4x = 2x, so 2x = 0.
Every Boolean ring is commutative.
Expand (x + y)2 = x2 + xy + yx + y2. Idempotence reduces this to x + y = x + xy + yx + y, so xy + yx = 0. Since the characteristic is 2 this gives xy = yx.
Adding a single identity to the ring axioms forces both commutativity and characteristic 2. This is an unusually strong consequence, and it explains why Boolean rings are so rigid.
A variety
Boolean rings are defined by the ring identities plus x2 ≈ x, so they form a variety. Closure under H, S and P follows, and free Boolean rings exist.
| Ring | Operations |
|---|---|
| Z/2Z | Ordinary arithmetic mod 2 |
| Su(X) with symmetric difference and intersection | a + b = symmetric difference; ab = intersection |
| Any product of copies of Z/2Z | Coordinatewise |
| Continuous functions from a Boolean space to Z/2Z | Pointwise |
The power set becomes a Boolean ring with symmetric difference as addition and intersection as multiplication. Symmetric difference is the characteristic-2 addition, and the empty set is the zero. This is the concrete model to keep in mind.
Ideals and prime ideals
Because Boolean rings are commutative with unit, the standard ideal theory applies — and it simplifies dramatically.
In a Boolean ring, every prime ideal is maximal, and the quotient by a prime ideal is isomorphic to Z/2Z.
If P is prime, then for any x, from x(x − 1) = x2 − x = 0 ∈ P, primeness gives x ∈ P or x − 1 ∈ P. So the quotient has only two elements.
Prime ideals of a Boolean ring correspond to ultrafilters of the associated Boolean algebra, and the set of them carries a topology. The spectrum of a Boolean ring, in the sense of commutative algebra, is exactly the Stone space of the corresponding Boolean algebra.
Frequently asked questions
Is every commutative ring of characteristic 2 Boolean?
No — idempotence is a strictly stronger condition. The polynomial ring over Z/2Z has characteristic 2 but x² ≠ x.
Are Boolean rings Noetherian?
Only when finite. An infinite Boolean ring has an infinite strictly increasing chain of ideals, so it is not Noetherian.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section IV.2, book pages 136-138.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.