nLab
ideal

This entry discusseds the notion of ideal in fair generality. For an entry closeer to the standard notion see at ideal in a monoid.

Definitions

Ideals show up both in ring theory and in lattice theory. Actually, both of these can be slightly generalised:

In rings and other rigs

A left ideal in a ring (or even rig) $R$ is a subset $I$ of (the underlying set of) $R$ such that:

• $0\in I$;
• $x+y\in I$ whenever $x,y\in I$;
• $xy\in I$ whenever $y\in I$, regardless of whether $x\in I$.

A right ideal in $R$ is a subset $I$ such that:

• $0\in I$;
• $x+y\in I$ whenever $x,y\in I$;
• $xy\in I$ whenever $x\in I$.

A two-sided ideal in $R$ is a subset $I$ that is both a left and right ideal; that is:

• $0\in I$;
• $x+y\in I$ whenever $x\in I$ and $y\in I$;
• $xy\in I$ whenever $x\in I$ or $y\in I$.

This generalises to:

• $x_1+\cdots +x_n\in I$ whenever $x_k\in I$ for every $k$;
• $x_1\cdots x_n\in I$ whenever $x_k\in I$ for some $k$.

Notice that all three kinds of ideal are equivalent for a commutative ring.

In lattices and other prosets

An ideal in a lattice (or even proset) $L$ is a subset $I$ of (the underlying set of) $L$ such that:

• There is an element of $I$ (so that $I$ is inhabited);
• if $x,y\in I$, then $x,y\le z$ for some $z\in I$;
• if $x\in I$ and $y\le x$, then $y\in I$ too.

We can make this look more algebraic if $L$ is a (bounded) join-semilattice:

• $\perp \in I$;
• $x\vee y\in I$ if $x,y\in I$;
• $y\in I$ whenever $x\vee y\in I$.

If $L$ is indeed a lattice, then we can make this look just like the ring version:

• $\perp \in I$;
• $x\vee y\in I$ whenever $x,y\in I$;
• $x\wedge y\in I$ whenever $x\in I$.

The concept of ideal is dual to that of filter. A subset of $L$ that satisfies the first two of the three axioms for an ideal in a proset is precisely a directed subset of $L$; notice that this is weaker than being a sub-join-semilattice even if $L$ is a lattice.

In both at once

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.

A Boolean algebra is a rig in two different ways: as a distributive lattice and as a Boolean ring. Fortunately, these actually give the same concept of ideal.

In monoids

There is a notion of ideal in a monoid, or more generally in a monoid object in any monoidal category $C$, which generalises the notion of ideal in a ri(n)g or in a (semi)lattice. That is, if $C$ is Ab, then a monoid in $C$ is a ring; if $C$ is Ab Mon, then a monoid in $C$ is a rig; and a semilattice is a commutative idempotent monoid in Set. See ideal in a monoid.

In categories

More generally still, passing from monoids to their many-object version there is a notion of ideal in a category. This is usually called a sieve. See there.

Kinds of ideals

An ideal $I$ is proper if there exists an element $x$ such that $x\notin I$. In a rig, $I$ is proper iff $1\notin I$; in a (bounded) lattice, $I$ is proper iff $\top \notin I$. If instead $x\in I$ for every $x$ (which follows if $1\in I$ or $\top \in I$), we have the improper ideal.

An ideal $I$ is prime if it is proper and it satsfies a binary condition corresponding to the nullary condition that is properness:

• In a rig, $x\in I$ or $y\in I$ if $xy\in I$;
• In a proset, $x\in I$ or $y\in I$ if, for all $z$, $z\in I$ if $z\le x$ or $z\le y$.
• In a lattice (simplifying the proset version to look like the rig verison), $x\in I$ or $y\in I$ if $x\wedge y\in I$.

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.

That every ideal is contained in a prime ideal is a prime ideal theorem; that every ideal is contained in a maximal ideal is a maximal ideal theorem.

An ideal $I$ is principal if there exists an element $x\in I$ such that $y$ is a multiple of $x$ (in a rig) or $y\le q$ (in an ordered set) whenever $y\in I$; we say that $I$ is generated by $x$. Every element $x$ generates a unique principal ideal, the set of all multiples of $x$ (in a rig) or the downset of $x$ (in an an order). In the noncommutative case, ‘multiple’ should be interpreted in a left/right sense to match that of ‘ideal’.

More generally, the ideals form a Moore collection of subsets of $R$ or $L$, so we have an ideal generated by any subset.