If is interpreted as a Boolean ring, is a ring ideal.
If is interpreted as a Boolean lattice, is a lattice ideal.
Before proving this equivalence, we want to mention that a Boolean ring is equivalent to a Boolean lattice, and a more commonly used terminology is a Boolean algebra, which is also valid, as it is an algebra over the ring of integers. The standard way of characterizing the ring structure from the lattice structure is by defining (called the symmetric difference) and . From this, we can “solve” for in terms of and : .
First, suppose is an ideal of the “ring” . If , then . Suppose now that and with . Then as well. So is a lattice ideal of .
Next, suppose is an ideal of the “lattice” . If , then both and are in since the first one is less than or equal to and the second less than or equal to , so their join is in as well, this means that . Furthermore, if and , then as well. As a result, is a ring ideal of . ∎
A subset of a Boolean algebra satisfying the two equivalent conditions above is called a Boolean ideal, or an ideal for short. A prime Boolean ideal is a prime lattice ideal, and a maximal Boolean ideal is a maximal lattice ideal. Again, these notions and their ring theoretic counterparts match exactly. In fact, one can say more about these ideals in the case of a Boolean algebra: prime ideals are precisely the maximal ideals. If is a Boolean ring, and is a maximal ideal of , then is isomorphic to .
Remark. The dual notion of a Boolean ideal is a Boolean filter, or a filter for short. A Boolean filter is just a lattice filter of the Boolean algebra when considered as a lattice. To see the connection between a Boolean ideal and a Boolean lattice, let us define, for any subset of a Boolean algebra , the set . It is easy to see that . Now, if is an ideal, then is a filter. Conversely, if is a filter, is a ideal. In fact, given any Boolean algebra, there is a Galois connection
|Date of creation||2013-03-22 17:01:59|
|Last modified on||2013-03-22 17:01:59|
|Last modified by||CWoo (3771)|
|Defines||prime Boolean ideal|
|Defines||maximal Boolean ideal|