PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (16)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Medium
[parent] lattice of ideals (Definition)

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=\lbrace J_i\mid i\in I\rbrace$ of (left) ideals of $ R$ ($ I$ is an index set), define
$\displaystyle \bigwedge S:=\bigcap S$   and$\displaystyle \qquad\bigvee S=\sum_i J_i,$
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
  • 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.
  • 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. $ \qedsymbol$

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

  • the “ideal” of finite sums of the elements of $ J_i$, or
  • the “ideal” generated by (elements of) $ J_i$, or
  • 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. $ \qedsymbol$

Remarks.



"lattice of ideals" is owned by CWoo.
(view preamble)

View style:

See Also: sum of ideals, lattice ideal, ideal completion of a poset


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: dcpo, order ideals, bottom, upper semilattice, lattice ideals, theory, quantale, structure, commutative, right, two-sided ideals, even, right ideals, algebraic, compact, principal ideals, ideal generated bies, algebraic lattice, compact elements, finitely generated, sublattice, join, generated by, proof, complete, operations, lattice, well-defined, implies, components, inverse, additive, sums, infinite, finite, contained, least upper bound, greatest lower bound, sum of ideals, index set, ideals, collection, complete lattice, partially ordered set, inclusion, order, left ideals, ring
There are 3 references to this entry.

This is version 10 of lattice of ideals, born on 2007-04-27, modified 2007-07-25.
Object id is 9275, canonical name is LatticeOfIdeals.
Accessed 876 times total.

Classification:
AMS MSC: 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
 11N80 (Number theory :: Multiplicative number theory :: Generalized primes and integers)
 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous)
 06B35 (Order, lattices, ordered algebraic structures :: Lattices :: Continuous lattices and posets, applications)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)