This entry discusseds the notion of ideal in fair generality. For an entry closeer to the standard notion see at ideal in a monoid.
A right ideal in is a subset such that:
A two-sided ideal in is a subset that is both a left and right ideal; that is:
This generalises to:
Notice that all three kinds of ideal are equivalent for a commutative ring.
We can make this look more algebraic if is a (bounded) join-semilattice:
If is indeed a lattice, then we can make this look just like the ring version:
The concept of ideal is dual to that of filter. A subset of that satisfies the first two of the three axioms for an ideal in a proset is precisely a directed subset of ; notice that this is weaker than being a sub-join-semilattice even if is a lattice.
A distributive lattice is both a lattice and a commutative rig; the two concepts of ideal are the same, as can be seen by comparing the definition for rigs to the last definition for lattices.
There is a notion of ideal in a monoid, or more generally in a monoid object in any monoidal category , which generalises the notion of ideal in a ri(n)g or in a (semi)lattice. That is, if is Ab, then a monoid in is a ring; if is Ab Mon, then a monoid in is a rig; and a semilattice is a commutative idempotent monoid in Set. See ideal in a monoid.
An ideal is proper if there exists an element such that . In a rig, is proper iff ; in a (bounded) lattice, is proper iff . If instead for every (which follows if or ), we have the improper ideal.
An ideal is prime if it is proper and it satsfies a binary condition corresponding to the nullary condition that is properness:
An ideal is maximal if it is maximal among proper ideals. A maximal ideal in a rig (including in a distributive lattice, but not in every lattice) is necessarily prime; a prime ideal in a Boolean algebra is necessarily maximal.
An ideal is principal if there exists an element such that is a multiple of (in a rig) or (in an ordered set) whenever ; we say that is generated by . Every element generates a unique principal ideal, the set of all multiples of (in a rig) or the downset of (in an an order). In the noncommutative case, ‘multiple’ should be interpreted in a left/right sense to match that of ‘ideal’.