ideal

This entry discusseds the notion of

idealin fair generality. For an entry closeer to the standard notion see atideal in a monoid.

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

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 $x, y \in I$;
- $x y \in I$ 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 $x, y \in I$;
- $x y \in I$ 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 $y \in I$;
- $x y \in I$ 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.

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 $x, y \leq z$ for some $z \in I$;
- if $x \in I$ and $y \leq x$, then $y \in I$ too.

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

- $\bot \in I$;
- $x \vee y \in I$ 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:

- $\bot \in I$;
- $x \vee y \in I$ 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.

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.

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.

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.

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 $x y \in I$;
- In a proset, $x \in I$ or $y \in I$ if, for all $z$, $z \in I$ if $z \leq x$ or $z \leq y$.
- 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 $y \leq q$ (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.

Revised on April 29, 2013 20:15:54
by Urs Schreiber
(89.204.138.79)