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ideal

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Contents

Definitions

Ideals show up both in ring theory and in lattice theory. We recall both of these below and look at some slight generalizations.

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.

Remarks
  • A left ideal in a ring RR may be equivalently defined as an RR-submodule of RR, viewing the latter as a left RR-module. Mutatis mutandis for right ideals.
  • A two-sided ideal in a ring RR may be equivalently defined as a sub-bimodule of RR, viewing the latter as an RR-bimodule.
  • The preceding remarks apply to rigs as well.
  • Considering the category of rings as a Barr-exact category, there is a natural bijection between congruence relations on a ring RR (internal to the category of rings) and two-sided ideals of RR; this associates to each ideal II the relation I\sim_I where x Iyx \sim_I y means xyIx - y \in I. This observation does not apply to the category of rigs.

This definition also makes sense for nonassociative rings (or rigs), with the left/right/two-sided coincidence in the commutative case.

In the case of a nonunital ring (or rig), if RR is being thought of as a module over some other ring KK, then it won't immediately follow that II is a submodule of RR over KK, and so one usually includes that requirement as well:

  • axIa x \in I whenever aKa \in K and xIx \in I

in the case of left-modules, and similarly in the case of right modules or bimodules. (Technically, this should be distinguished in the terminology, say by calling II an ideal of RR over KK or the like.) In particular, a (real or complex) Lie ideal of a Lie algebra RR is an ideal of RR over the real or complex field KK.

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

There are some common situations where these two kinds of ideal might seem to clash but fortunately do not:

  • 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 both a distributive lattice and a Boolean ring; again, the two concepts of ideal are the same (partly because the multiplication operators are the same, although there is still some checking to do regarding closure under addition).

On the other hand, every poset is a poset in an opposite way, and this does not give the same concept of ideal; an ideal in one is a filter in the opposite one. We are lucky that the convention for interpreting a Boolean ring as a lattice goes in the correct direction, or the two notions of ideal in a Boolean algebra would not match; or perhaps it is not a matter of luck, but the convention for which way to define ideals in a lattice was chosen precisely to match the conventions for Boolean algebras!

In monoids

There is a notion of ideal in a monoid (or even semigroup), 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. A left (right) ideal in a monoid (or semigroup) is a subset closed under multiplications with arbitrary elements in the semigroup from the left (right). See ideal in a monoid.

This generalizes all of the above notions of ideal except for ideals in prosets that are not (possibly unbounded) join-semilattices.

In categories

More generally still, passing from monoids to their many-object version a right ideal or sieve in a category is a subcategory closed under precomposition with morphisms from the entire category, and a cosieve or left ideal is the dual notion closed under postcompositions with morphisms in the entire category. See sieve.

In additive categories

In ringoids (small additive categories) and additive categories in general, a left/right/2-sided ideal is just the ideal (sieve/cosieve) in the sense of categories which is also closed under group operations in hom groups.

Kinds of ideals

Ideals form complete lattices where arbitrary meets are given by set-theoretic intersection. In other words, ideals form a Moore collection of subsets of RR if RR is a rig, or of LL if LL is a lattice. This implies we have an ideal generated by any subset: the intersection of all ideals containing the subset. A subset SS that generates a given ideal II may be called a subbase of II; then SS is a base if every element of II is a multiple (in a rig) or a predecessor (in an order) of some element of SS. (In particular, every singleton subset is a base of its generated ideal.) See also filter base and dualize for more about bases and subbases of ideals in lattices and other posets.

Certain kinds of ideals are often characterized by the roles they play in ideal lattices, or in terms of the Moore closure operator. Some examples follow.

The top element of an ideal lattice is called the improper ideal. That is to say, an ideal II is the improper ideal if xIx \in I for every xx (which follows if 1I1 \in I for the case of rigs, or I\top \in I for the case of bounded lattices). An ideal II is proper if it is not the improper ideal: if there exists an element xx such that xIx \notin I. So in a rig, II is proper iff 1I1 \notin I; in a (bounded) lattice, II is proper iff I\top \notin I.

An ideal is a maximal ideal 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; conversely, a prime ideal in a Boolean algebra is necessarily maximal.

An ideal II is principal if it is generated by a singleton. This means there exists an element xIx \in I such that yy is a multiple of xx (in a rig) or yxy \leq x (in an ordered set) whenever yIy \in I; we say that II is generated by xx. Thus every element xx generates a unique principal ideal, the set of all left/right/two-sided multiples of xx (axa x, xbx b, or axba x b if we are talking about left/right/two-sided ideals in a rig) or the downset of xx (in an an order). Clearly, every ideal II is a join over all the principal ideals P xP_x generated by the elements xx of II.

As discussed at ideals in a monoid, there is for two-sided ideals an operation of ideal multiplication, making the ideal lattice a quantale (cf. Day convolution). Namely, if I,JI, J are ideals, then their product IJI J is the ideal generated by all products xyx y with xI,yJx \in I, y \in J in the case of rigs. Similarly, in the case of lattices, we could define IJI J to be the ideal generated by all meets xyx \wedge y – but in this case the result is the same as IJI \cap J. In any case, we say that a proper ideal PP is prime if for any ideals I,JI, J, the condition IJPI J \subseteq P implies IPI \subseteq P or JPJ \subseteq P. In order theory, the term ‘prime ideal’ is usually used for a strongly irreducible ideal, since the two are equivalent for prosets.

An (left, right, or two-sided) ideal II is irreducible if it is proper and, whenever it is written as the intersection of two ideals (of the same kind) I=JKI = J \cap K, then at least one of JJ and KK equals II. If we replace equality here by containment, then we get something more analogous to the definition of prime ideal, called a strongly irreducible ideal. (Indeed, for prosets, prime ideals and strongly irreducible ideals are the same.) In general, prime ideals are strongly irreducible, and strongly irreducible ideals are irreducible, but the converses fail. An ideal is completely irreducible if whenever II is an intersection of any (possibly infinite) collection of ideals, II is equal to at least one of them. (This is not analogous to completely prime ideals.)

An ideal II is a nil ideal if for every xIx\in I, there is an nn\in \mathbb{N} with x n=0x^n=0. That is, every element of the ideal is nilpotent. If on the other hand, there is an nn\in \mathbb{N}, such that for every xIx\in I, x n=0x^n=0, the ideal is called nilpotent.

Proposition

A maximal ideal MM is prime.

Proof

Because the ideal lattice is a quantale, multiplication of ideals distributes over ideal joins. Suppose IJMI J \subseteq M for two ideals I,JI, J. If neither is contained in MM, then IM==JMI \vee M = \top = J \vee M (the improper ideal) since MM is maximal. Then

==(IM)(JM)=IJMJIMMM\top = \top \cdot \top = (I \vee M) \cdot (J \vee M) = I J \vee M J \vee I M \vee M M

where all four summands are contained in MM (IJMI J \subseteq M by supposition, and the other containments hold since MM is an ideal). Thus their join is contained in MM, so we have proved M\top \subseteq M, contradiction.

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.

Last revised on September 7, 2018 at 11:10:21. See the history of this page for a list of all contributions to it.