lattice of ideals

Let R be a ring. Consider the set L(R) of all left idealsMathworldPlanetmathPlanetmath of R. Order this set by inclusion, and we have a partially ordered setMathworldPlanetmath. In fact, we have the following:

Proposition 1.

L(R) is a complete latticeMathworldPlanetmath.


For any collectionMathworldPlanetmath S={JiiI} of (left) ideals of R (I is an index setMathworldPlanetmathPlanetmath), define

S:=S  and  S=iJi,

the sum of ideals Ji. We assert that S is the greatest lower boundMathworldPlanetmath of the Ji, and S the least upper bound of the Ji, and we show these facts separately

  • First, S is a left ideal of R: if a,bS, then a,bJi for all iI. Consequently, a-bJi and so a-bS. Furthermore, if rR, then raJi for any iI, so raS also. Hence S is a left ideal. By construction, S is clearly contained in all of Ji, and is clearly the largest such ideal.

  • For the second part, we want to show that S actually exists for arbitrary S. We know the existence of S if S is finite. Suppose now S is infiniteMathworldPlanetmath. Define J to be the set of finite sums of elements of iJi. If a,bJ, then a+b, being a finite sum itself, clearly belongs to J. Also, -aJ as well, since the additive inverse of each of the additivePlanetmathPlanetmath components of a is an element of iJi. Now, if rR, then raJ too, since multiplying each additive component of a by r (on the left) lands back in iJi. So J is a left ideal. It is evident that JiJ. Also, if M is a left ideal containing each Ji, then any finite sum of elements of Ji must also be in M, hence JM. This implies that J is the smallest ideal containing each of the Ji. Therefore S exists and is equal to J.

In summary, both S and S are well-defined, and exist for finite S, so L(R) is a latticeMathworldPlanetmathPlanetmath. Additionally, both operationsMathworldPlanetmath work for arbitrary S, so L(R) is completePlanetmathPlanetmathPlanetmathPlanetmath. ∎

From the above proof, we see that the sum S of ideals Ji can be equivalently interpreted as

  • the “ideal” of finite sums of the elements of Ji, or

  • the “ideal” generated by (elements of) Ji, or

  • the join of ideals Ji.

A special sublattice of L(R) is the lattice of finitely generatedMathworldPlanetmathPlanetmath ideals of R. It is not hard to see that this sublattice comprises precisely the compact elements in L(R).

Looking more closely at the above proof, we also have the following:

Corollary 1.

L(R) is an algebraic lattice.


As we have already shown, L(R) is a complete lattice. If J is any (left) ideal of R, by the previous remark, each J is the sum (or join) of ideals generated byPlanetmathPlanetmath individual elements of J. Since these ideals are principal idealsMathworldPlanetmathPlanetmath (generated by a single element), they are compact, and therefore L(R) is algebraicMathworldPlanetmath. ∎


  • One can easily reconstruct all of the above, if L(R) is the set of right ideals, or even two-sided ideals of R. We may distinguish the three notions: l.L(R),r.L(R), and L(R) as the lattices of left, right, and two-sided ideals of R.

  • When R is commutativePlanetmathPlanetmathPlanetmath, l.L(R)=r.L(R)=L(R). Furthermore, it can also be shown that L(R) has the additional structureMathworldPlanetmath of a quantale.

  • There is also a related result on lattice theory: the set Id(L) of lattice ideals in a upper semilatticePlanetmathPlanetmath L with bottom 0 forms a complete lattice. For a proof of this, see this entry (

  • However, the more general case is not true: the set of order ideals in a poset is a dcpo.

Title lattice of ideals
Canonical name LatticeOfIdeals
Date of creation 2013-03-22 16:59:40
Last modified on 2013-03-22 16:59:40
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 06B35
Classification msc 14K99
Classification msc 16D25
Classification msc 11N80
Classification msc 13A15
Related topic SumOfIdeals
Related topic LatticeIdeal
Related topic IdealCompletionOfAPoset