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.

Contents

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) RR is a subset II of (the underlying set of) RR such that:

  • 0I0 \in I;
  • x+yIx + y \in I whenever x,yIx, y \in I;
  • xyIx y \in I whenever yIy \in I, regardless of whether xIx \in I.

A right ideal in RR is a subset II such that:

  • 0I0 \in I;
  • x+yIx + y \in I whenever x,yIx, y \in I;
  • xyIx y \in I whenever xIx \in I.

A two-sided ideal in RR is a subset II that is both a left and right ideal; that is:

  • 0I0 \in I;
  • x+yIx + y \in I whenever xIx \in I and yIy \in I;
  • xyIx y \in I whenever xIx \in I or yIy \in I.

This generalises to:

  • x 1++x nIx_1 + \cdots + x_n \in I whenever x kIx_k \in I for every kk;
  • x 1x nIx_1 \cdots x_n \in I whenever x kIx_k \in I for some kk.

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) LL is a subset II of (the underlying set of) LL such that:

  • There is an element of II (so that II is inhabited);
  • if x,yIx, y \in I, then x,yzx, y \leq z for some zIz \in I;
  • if xIx \in I and yxy \leq x, then yIy \in I too.

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

  • I\bot \in I;
  • xyIx \vee y \in I if x,yIx, y \in I;
  • yIy \in I whenever xyIx \vee y \in I.

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

  • I\bot \in I;
  • xyIx \vee y \in I whenever x,yIx, y \in I;
  • xyIx \wedge y \in I whenever xIx \in I.

The concept of ideal is dual to that of filter. A subset of LL that satisfies the first two of the three axioms for an ideal in a proset is precisely a directed subset of LL; notice that this is weaker than being a sub-join-semilattice even if LL 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 CC, which generalises the notion of ideal in a ri(n)g or in a (semi)lattice. That is, if CC is Ab, then a monoid in CC is a ring; if CC is Ab Mon, then a monoid in CC 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 II is proper if there exists an element xx such that xIx \notin I. In a rig, II is proper iff 1I1 \notin I; in a (bounded) lattice, II is proper iff I\top \notin I. If instead xIx \in I for every xx (which follows if 1I1 \in I or I\top \in I), we have the improper ideal.

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

  • In a rig, xIx \in I or yIy \in I if xyIx y \in I;
  • In a proset, xIx \in I or yIy \in I if, for all zz, zIz \in I if zxz \leq x or zyz \leq y.
  • In a lattice (simplifying the proset version to look like the rig verison), xIx \in I or yIy \in I if xyIx \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 II is principal if there exists an element xIx \in I such that yy is a multiple of xx (in a rig) or yqy \leq q (in an ordered set) whenever yIy \in I; we say that II is generated by xx. Every element xx generates a unique principal ideal, the set of all multiples of xx (in a rig) or the downset of xx (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 RR or LL, so we have an ideal generated by any subset.

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