aarjaneiro/planetmath huggingface dataset:
ideal of an algebra

ideal of an algebra

\PMlinkescapephrase

left ideal

\PMlinkescapephrase

right ideal

\PMlinkescapephrase

two-sided ideal

Let

A

be an algebra over a ring

R

.

Definition -

A

left ideal

of

A

is a

subalgebra

I

⊆

A

such that

a

⁢

x

∈

I

whenever

a

∈

A

and

x

∈

I

.

Equivalently, a left ideal of

A

is a subset

I

⊂

A

such that

1.

x

-

y

∈

I

, for all

x

,

y

∈

I

.

2.

r

⁢

x

∈

I

, for all

r

∈

R

and

x

∈

I

.

3.

a

⁢

x

∈

I

, for all

a

∈

A

and

x

∈

I

Similarly one can define a

right ideal

by replacing condition 3 by:

x

⁢

a

∈

I

whenever

a

∈

A

and

x

∈

I

.

A

two-sided ideal

of

A

is a left ideal which is also a right ideal. Usually the word ”” by itself means two-sided ideal. Of course, all these notions coincide when

A

is

commutative

.

0.0.1

Remark

Since an algebra is also a ring, one might think of borrowing the definition of ideal from ring . The problem is that condition 2 would not be in general satisfied (unless the algebra is unital).

Title

ideal of an algebra

Canonical name

IdealOfAnAlgebra

Date of creation

2013-03-22 18:09:00

Last modified on

2013-03-22 18:09:00

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 16D25

Synonym

left ideal of an algebra

Synonym

right ideal of an algebra

Synonym

two-sided ideal of an algebra

Generated on Fri Feb 9 18:37:17 2018 by

LaTeXML
internal direct sum of ideals

internal direct sum of ideals

Let

R

be a ring and

𝔞

1

,

𝔞

2

, …,

𝔞

n

its ideals (left, right or two-sided).  We say that

R

is the

internal

direct sum

of these ideals, denoted by

R

=

𝔞

1

⊕

𝔞

2

⊕

⋯

⊕

𝔞

n

,

if both of the following conditions are true:

R

=

𝔞

1

+

𝔞

2

+

⋯

+

𝔞

n

,

𝔞

i

∩

∑

j

≠

i

𝔞

j

=

{

0

}

∀

i

.

Theorem

.

If

𝔞

1

,

𝔞

2

, …,

𝔞

n

are ideals of the ring

R

, then the following two statements are

equivalent

:

•

R

=

𝔞

1

⊕

𝔞

2

⊕

⋯

⊕

𝔞

n

.

•

Every

element

r

of

R

has a unique

expression

r

=

a

1

+

a

2

+

⋯

+

a

n

with

a

i

∈

𝔞

i

⁢

∀

i

.

Title

internal direct sum of ideals

Canonical

name

InternalDirectSumOfIdeals

Date of creation

2013-03-22 14:49:32

Last modified on

2013-03-22 14:49:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 16D25

Classification

msc 11N80

Classification

msc 13A15

Defines

internal direct sum of ideals

Generated on Fri Feb 9 18:37:19 2018 by

LaTeXML
large ideal

large ideal

An ideal

I

of a ring

R

is called a

large ideal

if for every ideal

J

of

R

such that

J

≠

{

0

}

,

I

∩

J

≠

{

0

}

A ring is

semiprime

iff every large ideal is dense.

Obviously all nontrivial ideal of an

integral domain

is a large ideal, and the

maximal ideal

of any non-trivial

local ring

is a large ideal.

References

1

N.J. Fine, L. Gillman, J. Lambek

,
”

Rings of Quotients

of Rings of Functions”,

Transcribed and edited into PDF from the original 1966 McGill University Press book

(see

http://tinyurl.com/24unqs

here, Editors: M. Barr, R. Raphael),

http://tinyurl.com/ytw3tj

Online download,
Accessed 24.10.2007

Title

large ideal

Canonical name

LargeIdeal

Date of creation

2013-03-22 15:37:28

Last modified on

2013-03-22 15:37:28

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

12

Author

jocaps (12118)

Entry type

Definition

Classification

msc 16D25

Generated on Fri Feb 9 18:37:21 2018 by

LaTeXML
maximal ideal is prime

maximal ideal is prime

Theorem.

In a

commutative ring

with

non-zero unity

, any

maximal ideal

is a

prime ideal

.

Proof.

Let

𝔪

be a maximal ideal of such a ring

R

and let the

ring product

r

⁢

s

belong to

𝔪

but e.g.

r

∉

𝔪

. The maximality of

𝔪

implies that

𝔪

+

(

r

)

=

R

=

(

1

)

.  Thus there exists an element

m

∈

𝔪

and an element

x

∈

R

such that

m

+

x

⁢

r

=

1

.  Now

m

and

r

⁢

s

belong to

𝔪

, whence

s

=

1

⁢

s

=

(

m

+

x

⁢

r

)

⁢

s

=

s

⁢

m

+

x

⁢

(

r

⁢

s

)

∈

𝔪

.

So we can say that along with

r

⁢

s

, at least one of its factors (

http://planetmath.org/

Product

) belongs to

𝔪

, and therefore

𝔪

is a prime ideal of

R

.

Title

maximal ideal is prime

Canonical name

MaximalIdealIsPrime

Date of creation

2013-03-22 17:37:59

Last modified on

2013-03-22 17:37:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16D25

Classification

msc 13A15

Related topic

SumOfIdeals

Related topic

MaximumIdealIsPrimeGeneralCase

Related topic

CriterionForMaximalIdeal

Generated on Fri Feb 9 18:37:23 2018 by

LaTeXML
maximal ideal is prime (general case)

maximal ideal is prime (general case)

Theorem.

In a ring (not necessarily

commutative

) with unity, any

maximal ideal

is a

prime ideal

.

Proof.

Let

𝔪

be a maximal ideal of such a ring

R

and suppose

R

has ideals

𝔞

and

𝔟

with

𝔞

⁢

𝔟

⊆

𝔪

,
but

𝔞

⊈

𝔪

.
Since

𝔪

is

maximal

, we must have

𝔞

+

𝔪

=

R

.
Then,

𝔟

=

R

⁢

𝔟

=

(

𝔞

+

𝔪

)

⁢

𝔟

=

𝔞

⁢

𝔟

+

𝔪

⁢

𝔟

⊆

𝔪

+

𝔪

=

𝔪

.

Thus, either

𝔞

⊆

𝔪

or

𝔟

⊆

𝔪

. This demonstrates that

𝔪

is prime.

Note that the condition that

R

has an

identity element

is essential. For otherwise, we may take

R

to be a finite

zero ring

. Such rings contain no proper prime ideals. As long as the number of elements of

R

is not prime,

R

will have a non-zero maximal ideal.

Title

maximal ideal is prime

(general case)

Canonical name

MaximalIdealIsPrimegeneralCase

Date of creation

2013-03-22 17:38:02

Last modified on

2013-03-22 17:38:02

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Theorem

Classification

msc 16D25

Related topic

MaximalIdealIsPrime

Generated on Fri Feb 9 18:37:25 2018 by

LaTeXML
modular ideal

modular ideal

Let

R

be a ring. A

left ideal

I

of

R

is said to be

modular

if there is an

e

∈

R

such that

r

⁢

e

-

r

∈

I

for all

r

∈

R

. In other words,

e

acts as a

right identity

element modulo

I

:

r

⁢

e

≡

r

(

mod

I

)

.

A right

modular ideal

is defined similarly, with

e

be a left identity modulo

I

.

Remark

. If an ideal

I

is modular both as a left ideal as well as a

right ideal

in

R

, then

R

/

I

is a

unital ring

. Furthermore, every (left, right, two-sided) ideal in a unital ring is modular, implying that the notion of modular ideals is only interesting in rings without

1

.

References

1

P. M. Cohn,

Further

Algebra

and Applications

, Springer (2003).

Title

modular ideal

Canonical name

ModularIdeal

Date of creation

2013-03-22 17:31:47

Last modified on

2013-03-22 17:31:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D25

Generated on Fri Feb 9 18:37:37 2018 by

LaTeXML
nilpotent ideal

nilpotent ideal

A left (right) ideal

I

of a ring

R

is a

nilpotent ideal

if

I

n

=

0

for some positive integer

n

. Here

I

n

denotes a

product of ideals

–

I

⋅

I

⁢

⋯

⁢

I

.

Title

nilpotent ideal

Canonical name

NilpotentIdeal

Date of creation

2013-03-22 12:01:27

Last modified on

2013-03-22 12:01:27

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D25

Generated on Fri Feb 9 18:37:39 2018 by

LaTeXML
PID

PID

A

principal ideal domain

is an

integral domain

where every
ideal is a

principal ideal

.

In a PID, an ideal

(

p

)

is maximal if and only if

p

is

irreducible

(and prime since any PID is also a UFD (

http://planetmath.org/PIDsAreUFDs

)).

Note that

subrings

of PIDs are not necessarily PIDs. (There is
an example of this within the entry

biquadratic field

.)

Title

PID

Canonical name

PID

Date of creation

2013-03-22 11:56:25

Last modified on

2013-03-22 11:56:25

Owner

mps (409)

Last modified by

mps (409)

Numerical id

13

Author

mps (409)

Entry type

Definition

Classification

msc 16D25

Classification

msc 13G05

Classification

msc 11N80

Classification

msc 13A15

Synonym

principal ideal domain

Related topic

UFD

Related topic

Irreducible

Related topic

Ideal

Related topic

IntegralDomain

Related topic

EuclideanRing

Related topic

EuclideanValuation

Related topic

ProofThatAnEuclideanDomainIsAPID

Related topic

WhyEuclideanDomains

Generated on Fri Feb 9 18:37:41 2018 by

LaTeXML
PID and UFD are equivalent in a Dedekind domain

PID and UFD are equivalent in a Dedekind domain

This article shows that if

A

is a

Dedekind domain

, then

A

is a UFD if and only if it is a PID. Note that this result implies the more specific result given in the article

unique factorization and ideals in ring of integers

.

Since any PID is a UFD, we need only prove the other direction. So assume

A

is a UFD, let

𝔭

be a nonzero (proper)

prime ideal

, and choose

0

≠

x

∈

𝔭

. Note that

x

is a nonunit since

𝔭

is a

proper ideal

. Since

A

is a UFD, we may write

x

uniquely (up to units) as

x

=

p

1

a

1

⁢

⋯

⁢

p

k

a

k

where the

p

i

are distinct

irreducibles

in

A

, the

a

i

are

positive integers

, and

k

>

0

since

x

is not a unit. Since

𝔭

is prime and

x

∈

𝔭

, it follows that some

p

i

, say

p

1

, is in

𝔭

. Then

(

p

1

)

⊂

𝔭

. But

(

p

1

)

is prime since clearly in a UFD any

ideal generated by

an irreducible is prime. Since

A

is Dedekind and thus has

Krull dimension

1, it must be that

(

p

1

)

=

𝔭

and thus

𝔭

is principal.

Title

PID and UFD are

equivalent

in a Dedekind domain

Canonical name

PIDAndUFDAreEquivalentInADedekindDomain

Date of creation

2013-03-22 17:53:45

Last modified on

2013-03-22 17:53:45

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 16D25

Classification

msc 13G05

Classification

msc 13A15

Classification

msc 11N80

Generated on Fri Feb 9 18:37:43 2018 by

LaTeXML
pairwise comaximal ideals property

pairwise comaximal ideals property

Proposition 1

.

Let

R

be a

commutative ring

with unity. For every pairwise

comaximal ideals

I

1

,

I

2

,

…

,

I

n

, the following holds:

I

1

∩

I

2

∩

…

∩

I

n

=

I

1

⁢

I

2

⁢

…

⁢

I

n

.

(1)

Proof.

We prove by

induction

on

n

. For

n

=

2

,

I

1

+

I

2

=

R

implies:

I

1

∩

I

2

=

R

⁢

(

I

1

∩

I

2

)

=

(

I

1

+

I

2

)

⁢

(

I

1

∩

I

2

)

⊆

I

1

⁢

I

2

.

(2)

The

converse

inclusion

is

trivial

. Assume now that the equality holds for

n

≥

2

:

J

:=

I

1

∩

I

2

∩

…

∩

I

n

=

I

1

⁢

I

2

⁢

…

⁢

I

n

. Since

I

n

+

1

+

I

j

=

R

, for every

j

≠

n

+

1

, there exist the

elements

a

j

∈

I

j

and

b

j

∈

I

n

+

1

such that

a

j

+

b

j

=

1

. The

product

c

:=

∏

j

=

1

n

a

j

=

∏

j

=

1

n

(

1

-

b

j

)

∈

1

+

I

n

+

1

. Also

c

∈

J

, then

1

∈

J

+

I

n

+

1

or

J

+

I

n

+

1

=

R

.

Applying the case

2

, the induction step is satisfied:

I

1

⁢

I

2

⁢

…

⁢

I

n

+

1

=

J

⁢

I

n

+

1

=

J

∩

I

n

+

1

=

I

1

∩

I

2

∩

…

∩

I

n

∩

I

n

+

1

.

(3)

∎

Title

pairwise comaximal ideals property

Canonical name

PairwiseComaximalIdealsProperty

Date of creation

2013-03-22 16:53:34

Last modified on

2013-03-22 16:53:34

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

9

Author

polarbear (3475)

Entry type

Result

Classification

msc 16D25

Generated on Fri Feb 9 18:37:45 2018 by

LaTeXML
primitive ideal

primitive ideal

Let

R

be a ring, and let

I

be an ideal of

R

.
We say that

I

is a

left (right)

primitive ideal

if there exists a simple left (right)

R

-module

X

such that

I

is the

annihilator

of

X

in

R

.

We say that

R

is a

left (right) primitive ring

if the

zero ideal

is a left (right) primitive ideal of

R

.

Note that

I

is a left (right) primitive ideal
if and only if

R

/

I

is a left (right) primitive ring.

Title

primitive ideal

Canonical name

PrimitiveIdeal

Date of creation

2013-03-22 12:01:45

Last modified on

2013-03-22 12:01:45

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D25

Synonym

primitive ring

Generated on Fri Feb 9 18:37:47 2018 by

LaTeXML
product of ideals

product of ideals

Let

R

be a ring, and let

A

and

B

be left (right) ideals of

R

. Then the

product

of the ideals

A

and

B

, which we denote

A

⁢

B

, is the left (right)

ideal generated by

all products

a

⁢

b

with

a

∈

A

and

b

∈

B

. Note that since sums of products of the form

a

⁢

b

with

a

∈

A

and

b

∈

B

are contained simultaneously in both

A

and

B

, we have

A

⁢

B

⊂

A

∩

B

.

Title

product of ideals

Canonical name

ProductOfIdeals

Date of creation

2013-03-22 11:50:59

Last modified on

2013-03-22 11:50:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16D25

Classification

msc 15A15

Classification

msc 46L87

Classification

msc 55U40

Classification

msc 55U35

Classification

msc 81R10

Classification

msc 46L05

Classification

msc 22A22

Classification

msc 81R50

Classification

msc 18B40

Related topic

SumOfIdeals

Related topic

QuotientOfIdeals

Related topic

PruferRing

Related topic

ProductOfLeftAndRightIdeal

Related topic

WellDefinednessOfProductOfFinitelyGeneratedIdeals

Generated on Fri Feb 9 18:37:48 2018 by

LaTeXML
product of left and right ideal

product of left and right ideal

Let

𝔞

and

𝔟

be ideals of a ring

R

.  Denote by

𝔞

⁢

𝔟

the subset of

R

formed by all finite sums of

products

a

⁢

b

with

a

∈

𝔞

and

b

∈

𝔟

.  It is straightforward to verify the following facts:

•

If

𝔞

is a left (

http://planetmath.org/Ideal

) and

𝔟

a

right ideal

,

𝔞

⁢

𝔟

is a

two-sided ideal

of

R

.

•

If both

𝔞

and

𝔟

are two-sided ideals, then

𝔞

⁢

𝔟

⊆

𝔞

∩

𝔟

.

Title

product of left and right ideal

Canonical name

ProductOfLeftAndRightIdeal

Date of creation

2013-03-22 17:38:09

Last modified on

2013-03-22 17:38:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16D25

Related topic

ProductOfIdeals

Related topic

Intersection

Related topic

IdealMultiplicationLaws

Generated on Fri Feb 9 18:37:50 2018 by

LaTeXML
proof of product of left and right ideal

proof of product of left and right ideal

Theorem 1

Let

a

and

b

be ideals of a ring

R

. Denote by

a

⁢

b

the subset of

R

formed by all finite sums of products

a

⁢

b

with

a

∈

a

and

b

∈

b

. Then if

a

is a left and

b

a

right ideal

,

a

⁢

b

is a

two-sided ideal

of

R

. If in

addition

both

a

and

b

are two-sided ideals, then

a

⁢

b

⊆

a

∩

b

.

Proof.

We must show that the

difference

of any two elements of

𝔞

⁢

𝔟

is in

𝔞

⁢

𝔟

, and that

𝔞

⁢

𝔟

is closed under

multiplication

by

R

. But both of these operations are linear in

𝔞

⁢

𝔟

; that is, if they hold for elements of the form

a

⁢

b

,

a

∈

𝔞

,

b

∈

𝔟

, then they hold for the general element of

𝔞

⁢

𝔟

. So we restrict our analysis to elements

a

⁢

b

.

Clearly if

a

1

,

a

2

∈

𝔞

,

b

1

,

b

2

∈

𝔟

, then

a

1

⁢

b

1

-

a

2

⁢

b

2

∈

𝔞

⁢

𝔟

by

definition

.

If

a

∈

𝔞

,

b

∈

𝔟

,

r

∈

R

, then

r

⋅

a

⁢

b

=

(

r

⋅

a

)

⁢

b

∈

𝔞

⁢

𝔟

⁢

since

⁢

𝔞

⁢

is a left ideal

a

⁢

b

⋅

r

=

a

⁢

(

b

⋅

r

)

∈

𝔞

⁢

𝔟

⁢

since

⁢

𝔟

⁢

is a right ideal

and thus

𝔞

⁢

𝔟

is a two-sided ideal. This proves the first statement.

If

𝔞

,

𝔟

are two-sided ideals, then

a

⁢

b

∈

𝔞

since

b

∈

R

; similarly,

a

⁢

b

∈

𝔟

. This proves the second statement.

Title

proof of product of left and right ideal

Canonical

name

ProofOfProductOfLeftAndRightIdeal

Date of creation

2013-03-22 17:41:25

Last modified on

2013-03-22 17:41:25

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Proof

Classification

msc 16D25

Generated on Fri Feb 9 18:37:52 2018 by

LaTeXML
proper ideal

proper ideal

Suppose

R

is a ring and

I

is an ideal of

R

. We say that

I

is a

proper ideal

if

I

is not equal to

R

.

Title

proper ideal

Canonical

name

ProperIdeal

Date of creation

2013-03-22 11:51:11

Last modified on

2013-03-22 11:51:11

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D25

Related topic

MaximalIdeal

Generated on Fri Feb 9 18:37:54 2018 by

LaTeXML
semiprime ideal

semiprime ideal

Let

R

be a ring.
An ideal

I

of

R

is a

semiprime ideal

if it

satisfies

the following

equivalent

conditions:

(a)

I

can be expressed as an

intersection

of

prime ideals

of

R

;

(b) if

x

∈

R

, and

x

⁢

R

⁢

x

⊂

I

, then

x

∈

I

;

(c) if

J

is a

two-sided ideal

of

R

and

J

2

⊂

I

, then

J

⊂

I

as well;

(d) if

J

is a

left ideal

of

R

and

J

2

⊂

I

, then

J

⊂

I

as well;

(e) if

J

is a

right ideal

of

R

and

J

2

⊂

I

, then

J

⊂

I

as well.

Here

J

2

is the

product of ideals

J

⋅

J

.

The ring

R

itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus

semiprime

.

A ring

R

is said to be a

semiprime ring

if its

zero ideal

is a semiprime ideal.

Note that an ideal

I

of

R

is semiprime if and only if the

quotient ring

R

/

I

is a semiprime ring.

Title

semiprime ideal

Canonical name

SemiprimeIdeal

Date of creation

2013-03-22 12:01:23

Last modified on

2013-03-22 12:01:23

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

11

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D25

Related topic

NSystem

Defines

semiprime ring

Defines

semiprime

Generated on Fri Feb 9 18:37:56 2018 by

LaTeXML
well-definedness of product of finitely generated ideals

well-definedness of product of finitely generated ideals

Ler

R

be of a

commutative ring

with

nonzero unity

.  If

𝔞

=

(

a

1

,

…

,

a

m

)

=

(

α

1

,

…

,

α

μ

)

(1)

and

𝔟

=

(

b

1

,

…

,

b

n

)

=

(

β

1

,

…

,

β

ν

)

(2)

are two

finitely generated

ideals of

R

, both with two , then the ideals

𝔠

:=

(

a

1

⁢

b

1

,

…

,

a

i

⁢

b

j

,

…

,

a

m

⁢

b

n

)

and

𝔡

:=

(

α

1

⁢

β

1

,

…

,

α

i

⁢

β

j

,

…

,

α

μ

⁢

β

ν

)

are equal.

Proof.

By (1) and (2), for every

i

,

j

, there are elements

r

i

⁢

k

,

s

j

⁢

l

of

R

such that

a

i

=

r

i

⁢

1

⁢

α

1

+

…

+

r

i

⁢

μ

⁢

α

μ

,

b

j

=

s

j

⁢

1

⁢

β

1

+

…

+

s

j

⁢

ν

⁢

β

ν

.

(3)

Multiplying the

equations

(3) we see that

a

i

⁢

b

j

=

(

r

i

⁢

1

⁢

s

j

⁢

1

)

⁢

(

α

1

⁢

β

1

)

+

(

r

i

⁢

2

⁢

s

j

⁢

1

)

⁢

(

α

2

⁢

β

1

)

+

…

+

(

r

i

⁢

μ

⁢

s

j

⁢

ν

)

⁢

(

α

μ

⁢

β

ν

)

,

whence the

generators

a

i

⁢

b

j

of

𝔠

belong to

𝔡

and consecuently

𝔠

⊆

𝔡

.  The reverse containment is seen similarly.

Title

well-definedness of product of finitely generated ideals

Canonical name

WelldefinednessOfProductOfFinitelyGeneratedIdeals

Date of creation

2013-03-22 19:12:56

Last modified on

2013-03-22 19:12:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16D25

Related topic

WellDefined

Related topic

ProductOfIdeals

Related topic

ProductOfFinitelyGeneratedIdeals

Generated on Fri Feb 9 18:37:58 2018 by

LaTeXML
zero ideal

zero ideal

In any ring, the set consisting only of the

zero element

(i.e. the

additive identity

) is an ideal of the left, right, and two-sided

varieties

. It is the smallest ideal in any ring.

Title

zero ideal

Canonical name

ZeroIdeal

Date of creation

2013-03-22 11:51:02

Last modified on

2013-03-22 11:51:02

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D25

Classification

msc 11A41

Generated on Fri Feb 9 18:38:00 2018 by

LaTeXML
absolutely flat

absolutely flat

A ring

A

is

absolutely flat

if every module over

A

is flat.

For

commutative rings

with unity, a ring is absolutely flat if and only if every

principal ideal

is

idempotent

.

Some properties:

•

Boolean rings

are flat.

•

Homomorphic images

of absolutely flat rings are flat.

•

Absolutely flat

local rings

are fields.

•

In absolutely flat rings, non-units are

zero divisors

.

References

1

Introduction

to

, by Atiyah and MacDonald.

Title

absolutely flat

Canonical name

AbsolutelyFlat

Date of creation

2013-03-22 14:35:52

Last modified on

2013-03-22 14:35:52

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16D40

Generated on Fri Feb 9 18:38:01 2018 by

LaTeXML
cardinalities of bases for modules

cardinalities of bases for modules

Let

R

be a ring and

M

a

left module

over

R

.

Proposition 1

.

If

M

has a finite basis, then all bases for

M

are finite.

Proof.

Suppose

A

=

{

a

1

,

…

,

a

n

}

is a finite basis for

M

, and

B

is another basis for

M

. Each

element

in

A

can be expressed as a finite

linear combination

of elements in

B

. Since

A

is finite, only a finite number of elements in

B

are needed to express elements of

A

. Let

C

=

{

b

1

,

…

,

b

m

}

be this finite subset (of

B

).

C

is

linearly independent

because

B

is. If

C

≠

B

, pick

b

∈

B

-

C

. Then

b

is

expressible

as a linear combination of elements of

A

, and subsequently a linear combination of elements of

C

. This means that

b

=

r

1

⁢

b

1

+

⋯

+

r

m

⁢

b

m

, or

0

=

-

b

+

r

1

⁢

b

1

+

⋯

⁢

r

m

⁢

b

m

, contradicting the linear independence of

C

.
∎

Proposition 2

.

If

M

has an

infinite

basis, then all bases for

M

have the same

cardinality

.

Proof.

Suppose

A

be a basis for

M

with

|

A

|

≥

ℵ

0

, the smallest infinite

cardinal

, and

B

is another basis for

M

. We want to show that

|

B

|

=

|

A

|

. First, notice that

|

B

|

≥

ℵ

0

by the previous

proposition

. Each element

a

∈

A

can be expressed as a

finite

linear combination of elements of

B

, so let

B

a

be the

collection

of these elements. Now,

B

a

is uniquely determined by

a

, as

B

is a basis. Also,

B

a

is finite. Let

B

′

=

⋃

a

∈

A

B

a

.

Since

A

spans

M

, so does

B

′

. If

B

′

≠

B

, pick

b

∈

B

-

B

′

, so that

b

is a linear combination of elements of

B

′

. Moving

b

to the other side of the expression and we have expressed

0

as a non-trivial linear combination of elements of

B

, contradicting the linear independence of

B

. Therefore

B

′

=

B

. This means

|

B

|

=

|

⋃

a

∈

A

B

a

|

≤

ℵ

0

⁢

|

A

|

=

|

A

|

.

Similarly, every element in

B

is expressible as a finite linear combination of elements in

A

, and using the same

argument

as above,

|

A

|

≤

ℵ

0

⁢

|

B

|

≤

|

B

|

.

By

Schroeder-Bernstein theorem

, the two

inequalities

can be combined to form the equality

|

A

|

=

|

B

|

.
∎

Title

cardinalities of bases for modules

Canonical name

CardinalitiesOfBasesForModules

Date of creation

2013-03-22 18:06:33

Last modified on

2013-03-22 18:06:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 16D40

Classification

msc 13C05

Classification

msc 15A03

Generated on Fri Feb 9 18:38:21 2018 by

LaTeXML