lattice of ideals

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

∣

i

∈

I

}

of (left) ideals of

R

(

I

is an

index set

), define

⋀

S

:=

⋂

S

and

⋁

S

=

∑

i

J

i

,

the

sum of ideals

J

i

. We assert that

⋀

S

is the

greatest lower bound

of the

J

i

, and

⋁

S

the least

upper bound

of the

J

i

, and we show these facts separately

•

First,

⋀

S

is a left ideal of

R

: if

a

,

b

∈

⋀

S

, then

a

,

b

∈

J

i

for all

i

∈

I

. Consequently,

a

-

b

∈

J

i

and so

a

-

b

∈

⋀

S

. Furthermore, if

r

∈

R

, then

r

⁢

a

∈

J

i

for any

i

∈

I

, so

r

⁢

a

∈

⋀

S

also. Hence

⋀

S

is a left ideal. By construction,

⋀

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

⋁

S

actually exists for arbitrary

S

. We know the

existence

of

⋁

S

if

S

is finite. Suppose now

S

is

infinite

. Define

J

to be the set of finite sums of elements of

⋃

i

J

i

. If

a

,

b

∈

J

, then

a

+

b

, being a finite sum itself, clearly belongs to

J

. Also,

-

a

∈

J

as well, since the

additive inverse

of each of the

additive

components

of

a

is an element of

⋃

i

J

i

. Now, if

r

∈

R

, then

r

⁢

a

∈

J

too, since multiplying each additive component of

a

by

r

(on the left) lands back in

⋃

i

J

i

. So

J

is a left ideal. It is evident that

J

i

⊆

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

⊆

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

⋁

S

and

⋀

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

•

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

.
∎

Remarks

.

•

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

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

.

•

There is also a related result on

lattice theory

: the set

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

).

•

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

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