lattice of ideals

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

Proposition 1.

$L(R)$ is a complete lattice.

Proof.

For any collection $S=\{J_{i}\mid i\in I\}$ of (left) ideals of $R$ ( $I$ is an index set), define

\bigwedge S:=\bigcap S\qquad\mbox{and}\qquad\bigvee S=\sum_{i}J_{i},

the sum of ideals $J_{i}$ . We assert that $\bigwedge S$ is the greatest lower bound of the $J_{i}$ , and $\bigvee S$ the least upper bound of the $J_{i}$ , and we show these facts separately

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First, $\bigwedge S$ is a left ideal of $R$ : if $a,b\in\bigwedge S$ , then $a,b\in J_{i}$ for all $i\in I$ . Consequently, $a-b\in J_{i}$ and so $a-b\in\bigwedge S$ . Furthermore, if $r\in R$ , then $ra\in J_{i}$ for any $i\in I$ , so $ra\in\bigwedge S$ also. Hence $\bigwedge S$ is a left ideal. By construction, $\bigwedge S$ is clearly contained in all of $J_{i}$ , and is clearly the largest such ideal.
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For the second part, we want to show that $\bigvee S$ actually exists for arbitrary $S$ . We know the existence of $\bigvee S$ if $S$ is finite. Suppose now $S$ is infinite. Define $J$ to be the set of finite sums of elements of $\bigcup_{i}J_{i}$ . If $a,b\in J$ , then $a+b$ , being a finite sum itself, clearly belongs to $J$ . Also, $-a\in J$ as well, since the additive inverse of each of the additive components of $a$ is an element of $\bigcup_{i}J_{i}$ . Now, if $r\in R$ , then $ra\in J$ too, since multiplying each additive component of $a$ by $r$ (on the left) lands back in $\bigcup_{i}J_{i}$ . So $J$ is a left ideal. It is evident that $J_{i}\subseteq J$ . Also, if $M$ is a left ideal containing each $J_{i}$ , then any finite sum of elements of $J_{i}$ must also be in $M$ , hence $J\subseteq M$ . This implies that $J$ is the smallest ideal containing each of the $J_{i}$ . Therefore $S$ exists and is equal to $J$ .

In summary, both $\bigvee S$ and $\bigwedge S$ are well-defined, and exist for finite $S$ , so $L(R)$ is a lattice. Additionally, both operations work for arbitrary $S$ , so $L(R)$ is complete. ∎

From the above proof, we see that the sum $S$ of ideals $J_{i}$ can be equivalently interpreted as

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the “ideal” of finite sums of the elements of $J_{i}$ , or
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the “ideal” generated by (elements of) $J_{i}$ , or
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the join of ideals $J_{i}$ .

A special sublattice of $L(R)$ is the lattice of finitely generated 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.

Proof.

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 by individual elements of $J$ . Since these ideals are principal ideals (generated by a single element), they are compact, and therefore $L(R)$ is algebraic. ∎

Remarks.

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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 $R$ .
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When $R$ is commutative, $l.L(R)=r.L(R)=L(R)$ . Furthermore, it can also be shown that $L(R)$ has the additional structure of a quantale.
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There is also a related result on lattice theory: the set $\operatorname{Id}(L)$ of lattice ideals in a upper semilattice $L$ with bottom $0$ forms a complete lattice. For a proof of this, see this entry (http://planetmath.org/IdealCompletionOfAPoset).
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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