aarjaneiro/planetmath huggingface dataset:
nil and nilpotent ideals

nil and nilpotent ideals

An

element

x

of a ring is

nilpotent

if

x

n

=

0

for some

positive integer

n

.

A ring

R

is

nil

if every element in

R

is nilpotent. Similarly, a one- or

two-sided ideal

is called

nil

if each of its elements is nilpotent.

A ring

R

[resp. a one- or two sided ideal

A

] is

nilpotent

if

R

n

=

0

[resp.

A

n

=

0

] for some positive integer

n

.

A ring or an ideal is

locally nilpotent

if every

finitely generated

subring

is nilpotent.

The following

implications

hold for rings (or ideals):

nilpotent

⇒

locally nilpotent

⇒

nil

Title

nil and nilpotent ideals

Canonical name

NilAndNilpotentIdeals

Date of creation

2013-03-22 13:13:25

Last modified on

2013-03-22 13:13:25

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 16N40

Related topic

KoetheConjecture

Defines

nil

Defines

nil ring

Defines

nil ideal

Defines

nil right ideal

Defines

nil left ideal

Defines

nil subring

Defines

nilpotent

Defines

nilpotent element

Defines

nilpotent ring

Defines

nilpotent ideal

Defines

nilpotent right ideal

Defines

nilpotent left ideal

Defines

nilpotent subring

Defines

locally nilpotent

Defines

locally nilpotent ring

Defines

locally nilpo

Generated on Fri Feb 9 18:41:31 2018 by

LaTeXML
nil is a radical property

nil is a radical property

We must show that the nil

property

,

𝒩

, is a

radical property

, that is that it

satisfies

the following conditions:

1.

The class of

𝒩

-rings is

closed under

homomorphic images

.

2.

Every ring

R

has a largest

𝒩

-ideal, which contains all other

𝒩

-ideals of

R

. This ideal is written

𝒩

⁢

(

R

)

.

3.

𝒩

⁢

(

R

/

𝒩

⁢

(

R

)

)

=

0

.

It is easy to see that the homomorphic image of a

nil ring

is nil, for if

f

:

R

→

S

is a

homomorphism

and

x

n

=

0

, then

f

⁢

(

x

)

n

=

f

⁢

(

x

n

)

=

0

.

The sum of all nil ideals is nil (see proof

http://planetmath.org/node/5650

here), so this sum is the largest nil ideal in the ring.

Finally, if

N

is the largest nil ideal in

R

, and

I

is an ideal of

R

containing

N

such that

I

/

N

is nil, then

I

is also nil (see proof

http://planetmath.org/node/5650

here). So

I

⊆

N

by

definition

of

N

. Thus

R

/

N

contains no nil ideals.

Title

nil is a radical property

Canonical

name

NilIsARadicalProperty

Date of creation

2013-03-22 14:12:58

Last modified on

2013-03-22 14:12:58

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Proof

Classification

msc 16N40

Related topic

PropertiesOfNilAndNilpotentIdeals

Generated on Fri Feb 9 18:41:33 2018 by

LaTeXML
nilpotency is not a radical property

nilpotency is not a radical property

Nilpotency is not a

radical property

, because a ring does not, in general, contain a largest

nilpotent ideal

.

Let

k

be a field, and let

S

=

k

⁢

[

X

1

,

X

2

,

…

]

be the ring of polynomials over

k

in infinitely many variables

X

1

,

X

2

,

…

.
Let

I

be the ideal of

S

generated by

{

X

n

n

+

1

∣

n

∈

(

N

)

}

.
Let

R

=

S

/

I

. Note that

R

is

commutative

.

For each

n

, let

A

n

=

∑

k

=

1

n

R

⁢

X

n

.
Let

A

=

⋃

A

n

=

∑

k

=

1

∞

R

⁢

X

n

.

Then each

A

n

is

nilpotent

, since it is the sum of finitely many nilpotent ideals (see proof

http://planetmath.org/node/5650

here). But

A

is nil, but not nilpotent. Indeed, for any

n

, there is an element

x

∈

A

such that

x

n

≠

0

, namely

x

=

X

n

, and so we cannot have

A

n

=

0

.

So

R

cannot have a largest nilpotent ideal, for this ideal would have to contain all the ideals

A

n

and therefore

A

.

Title

nilpotency is not a radical property

Canonical name

NilpotencyIsNotARadicalProperty

Date of creation

2013-03-22 14:13:02

Last modified on

2013-03-22 14:13:02

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

4

Author

mclase (549)

Entry type

Proof

Classification

msc 16N40

Generated on Fri Feb 9 18:41:35 2018 by

LaTeXML
upper nilradical

upper nilradical

The

upper nilradical

Nil

*

⁡

(

R

)

of

R

is the sum (

http://planetmath.org/SumOfIdeals

) of all (two-sided)

nil ideals

in

R

. In other words,

a

∈

N

⁢

i

⁢

l

*

⁢

R

iff

a

can be expressed as a (finite) sum of

nilpotent elements

.

It is not hard to see that

Nil

*

⁡

(

R

)

is the largest nil ideal in

R

. Furthermore, we have that

Nil

*

⁡

(

R

)

⊆

Nil

*

⁡

(

R

)

⊆

J

⁢

(

R

)

, where

Nil

*

⁡

(

R

)

is the lower

radical

or

prime radical

of

R

, and

J

⁢

(

R

)

is the

Jacobson radical

of

R

.

Remarks

.

•

If

R

is

commutative

, then

Nil

*

⁡

(

R

)

=

Nil

*

⁡

(

R

)

=

Nil

⁡

(

R

)

, the

nilradical

of

R

, consisting of all nilpotent elements.

•

If

R

is left (or right)

artinian

, then

Nil

*

⁡

(

R

)

=

Nil

*

⁡

(

R

)

=

J

⁢

(

R

)

.

Title

upper nilradical

Canonical name

UpperNilradical

Date of creation

2013-03-22 17:29:06

Last modified on

2013-03-22 17:29:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16N40

Generated on Fri Feb 9 18:41:47 2018 by

LaTeXML
reduced ring

reduced ring

A ring

R

is said to be a

reduced ring

if

R

contains no non-zero

nilpotent elements

. In other words,

r

2

=

0

implies

r

=

0

for any

r

∈

R

.

Below are some examples of reduced rings.

•

A reduced ring is

semiprime

.

•

A ring is a domain (

http://planetmath.org/CancellationRing

) iff it is prime (

http://planetmath.org/PrimeRing

) and

reduced

.

•

A

commutative

semiprime ring

is reduced. In particular, all

integral domains

and

Boolean rings

are reduced.

•

Assume that

R

is commutative, and let

A

be the set of all nilpotent elements. Then

A

is an ideal of

R

, and that

R

/

A

is reduced (for if

(

r

+

A

)

2

=

0

, then

r

2

∈

A

, so

r

2

, and consequently

r

, is

nilpotent

, or

r

∈

A

).

An example of a reduced ring with zero-divisors is

ℤ

n

, with

multiplication

defined componentwise:

(

a

1

,

…

,

a

n

)

⁢

(

b

1

,

…

,

b

n

)

:=

(

a

1

⁢

b

1

,

…

,

a

n

⁢

b

n

)

. A ring of

functions

taking values in a reduced ring is also reduced.

Some prototypical

examples of rings

that are not reduced are

ℤ

8

, since

4

2

=

0

, as well as any

matrix ring

over any ring; as illustrated by the

instance

below

(

0

1

0

0

)

⁢

(

0

1

0

0

)

=

(

0

0

0

0

)

.

Title

reduced ring

Canonical name

ReducedRing

Date of creation

2013-03-22 14:18:12

Last modified on

2013-03-22 14:18:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16N60

Synonym

nilpotent-free

Generated on Fri Feb 9 18:41:49 2018 by

LaTeXML
prime radical

prime radical

The

prime radical

or

lower nilradical

of a ring

R

is the intersection of all the

prime ideals

of

R

.

Note that the prime radical is the smallest

semiprime ideal

of

R

, and that

R

is a

semiprime ring

if and only if its prime radical is the

zero ideal

.

Title

prime radical

Canonical name

PrimeRadical

Date of creation

2013-03-22 12:01:29

Last modified on

2013-03-22 12:01:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16N80

Synonym

lower nilradical

Related topic

RadicalOfAnIdeal

Related topic

Nilradical

Generated on Fri Feb 9 18:41:51 2018 by

LaTeXML
radical theory

radical theory

Let

𝒳

represent

a property which a ring may or may not have. This property may be anything at all: what is important is that for any ring

R

, the statement “

R

has property

𝒳

” is either true or false.

We say that a ring which has the property

𝒳

is an

𝒳

-ring. An ideal

I

of a ring

R

is called an

𝒳

-ideal if, as a ring, it is an

𝒳

-ring. (Note that this definition only makes sense if rings are not required to have

identity elements

; otherwise and ideal is not, in general, a ring. Rings are not required to have an identity element in

radical theory

.)

The property

𝒳

is a

radical property

if it

satisfies

:

1.

The class of

𝒳

-rings is

closed under

homomorphic images

.

2.

Every ring

R

has a largest

𝒳

-ideal, which contains all other

𝒳

-ideals of

R

. This ideal is written

𝒳

⁢

(

R

)

.

3.

𝒳

⁢

(

R

/

𝒳

⁢

(

R

)

)

=

0

.

The ideal

𝒳

⁢

(

R

)

is called the

X

-

radical

of

R

. A ring is called

X

-radical

if

𝒳

⁢

(

R

)

=

R

, and is called

X

-

semisimple

if

𝒳

⁢

(

R

)

=

0

.

If

𝒳

is a radical property, then the class of

𝒳

-rings is also called the class of

X

-radical rings

.

The class of

𝒳

-radical rings is closed under

ideal extensions

. That is, if

A

is an ideal of

R

, and

A

and

R

/

A

are

𝒳

-radical, then so is

R

.

Radical theory

is the study of radical properties and their interrelations. There are several well-known radicals which are of independent interest in ring theory (See examples – to follow).

The class of all radicals is however very large. Indeed, it is possible to show that any

partition

of the class of

simple rings

into two classes

ℛ

and

𝒮

such that

isomorphic

simple rings are in the same class, gives rise to a radical

𝒳

with the property that all rings in

ℛ

are

𝒳

-radical and all rings in

𝒮

are

𝒳

-semisimple. In fact, there are at least two distinct radicals for each such partition.

A radical

𝒳

is

hereditary

if every ideal of an

𝒳

-radical ring is also

𝒳

-radical.

A radical

𝒳

is

supernilpotent

if the class of

𝒳

-rings contains all

nilpotent rings

.

1

Examples

Nil is a radical property. This property defines the nil radical,

𝒩

.

Nilpotency is not a radical property.

Quasi-regularity

is a radical property. The associated radical is the

Jacobson radical

,

𝒥

.

Title

radical theory

Canonical name

RadicalTheory

Date of creation

2013-03-22 13:13:02

Last modified on

2013-03-22 13:13:02

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

10

Author

mclase (549)

Entry type

Definition

Classification

msc 16N80

Related topic

JacobsonRadical

Defines

radical

Defines

radical property

Defines

semisimple

Defines

hereditary

Defines

hereditary radical

Defines

supernilpotent

Defines

supernilpotent radical

Generated on Fri Feb 9 18:41:53 2018 by

LaTeXML
an Artinian integral domain is a field

an Artinian integral domain is a field

Let

R

be an

integral domain

, and assume that

R

is

Artinian

.

Let

a

∈

R

with

a

≠

0

.
Then

R

⊇

a

⁢

R

⊇

a

2

⁢

R

⊇

⋯

.

As

R

is Artinian, there is some

n

∈

ℕ

such that

a

n

⁢

R

=

a

n

+

1

⁢

R

.
There exists

r

∈

R

such that

a

n

=

a

n

+

1

⁢

r

,
that is,

a

n

⁢

1

=

a

n

⁢

(

a

⁢

r

)

.
But

a

n

≠

0

(as

R

is an integral domain), so we have

1

=

a

⁢

r

.
Thus

a

is a unit.

Therefore, every Artinian integral domain is a field.

Title

an Artinian integral domain is a field

Canonical name

AnArtinianIntegralDomainIsAField

Date of creation

2013-03-22 12:49:37

Last modified on

2013-03-22 12:49:37

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Theorem

Classification

msc 16P20

Classification

msc 13G05

Related topic

AFiniteIntegralDomainIsAField

Generated on Fri Feb 9 18:41:54 2018 by

LaTeXML
Hopkins theorem

Hopkins theorem

Theorem 1

If a

ring with identity

is

left Artinian

then it is

left Noetherian

.

Title

Hopkins theorem

Canonical name

HopkinsTheorem

Date of creation

2013-03-22 16:03:53

Last modified on

2013-03-22 16:03:53

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 16P20

Classification

msc 16P40

Generated on Fri Feb 9 18:41:59 2018 by

LaTeXML
equivalent defining conditions on a Noetherian ring

equivalent defining conditions on a Noetherian ring

Let

R

be a ring. Then the following are

equivalent

:

1.

every

left ideal

of

R

is

finitely generated

,

2.

the

ascending chain condition

on left ideals holds in

R

,

3.

every non-empty family of left ideals has a

maximal element

.

Proof.

(

1

⇒

2

)

. Let

I

1

⊆

I

2

⊆

⋯

be an ascending chain of left ideals in

R

. Let

I

be the union of all

I

j

,

j

=

1

,

2

,

…

. Then

I

is a left ideal, and hence finitely generated, by, say,

a

1

,

⋯

⁢

a

n

. Now each

a

i

belongs to some

I

α

i

. Take the largest of these, say

I

α

k

. Then

a

i

∈

I

α

k

for all

i

=

1

,

…

,

n

, and therefore

I

⊆

I

α

k

. But

I

α

k

⊆

I

by the definition of

I

, the

equality

follows.

(

2

⇒

3

)

. Let

𝒮

be a non-empty family of left ideals in

R

. Since

𝒮

is non-empty, take any left ideal

I

1

∈

𝒮

. If

I

1

is

maximal

, then we are done. If not,

𝒮

-

{

I

1

}

must be non-empty, such that pick

I

2

from this

collection

so that

I

1

⊆

I

2

(we can find such

I

2

, for otherwise

I

1

would be maximal). If

I

2

is not maximal, pick

I

3

from

𝒮

-

{

I

1

,

I

2

}

such that

I

1

⊆

I

2

⊆

I

3

, and so on. By

assumption

, this can not go on indefinitely. So for some positive integer

n

, we have

I

n

=

I

m

for all

m

≥

n

, and

I

n

is our desired maximal element.

(

3

⇒

1

)

. Let

I

be a left ideal in

R

. Let

𝒮

be the family of all finitely generated ideals of

R

contained

in

I

.

𝒮

is non-empty since

(

0

)

is in it. By assumption

𝒮

has a maximal element

J

. If

J

≠

I

, then take an

element

a

∈

I

-

J

. Then

⟨

J

,

a

⟩

is finitely generated and contained in

I

, so an element of

𝒮

, contradicting the maximality of

J

. Hence

J

=

I

, in other words,

I

is finitely generated.
∎

A ring satisfying any, and hence all three, of the above conditions is defined to be a

left Noetherian ring

. A right Noetherian ring is similarly defined.

Title

equivalent defining conditions on a

Noetherian ring

Canonical name

EquivalentDefiningConditionsOnANoetherianRing

Date of creation

2013-03-22 18:04:27

Last modified on

2013-03-22 18:04:27

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 16P40

Generated on Fri Feb 9 18:42:08 2018 by

LaTeXML
Krull-Schmidt theorem

Krull-Schmidt theorem

A group

G

is said to

satisfy

the

ascending chain condition

(or ACC) on

normal subgroups

if there is no

infinite

ascending proper chain

G

1

⊊

G

2

⊊

G

3

⁢

⋯

with each

G

i

a normal subgroup of

G

.

Similarly,

G

is said to satisfy the

descending chain condition

(or DCC) on normal subgroups if there is no infinite descending proper chain of normal subgroups of

G

.

One can show that if a nontrivial group satisfies either the ACC or the DCC on normal subgroups, then that group can be expressed as the internal

direct product

of finitely many

indecomposable

subgroups

. If

both

the ACC and DCC are satisfied, the Krull-Schmidt theorem guarantees that this “decomposition into indecomposables” is

essentially unique

. (Note that every

finite group

satisfies both the ACC and DCC on normal subgroups.)

Krull-Schmidt theorem:

Let

G

be a nontrivial group satisfying both the ACC and DCC on its normal subgroups. Suppose

G

=

G

1

×

⋯

×

G

n

and

G

=

H

1

×

⋯

×

H

m

(internal direct products) where each

G

i

and

H

i

is indecomposable. Then

n

=

m

and, after reindexing,

G

i

≅

H

i

for each

i

. Moreover, for all

k

<

n

,

G

=

G

1

×

⋯

×

G

k

×

H

k

+

1

×

⋯

×

H

n

.

For proof, see Hungerford’s

Algebra

.

Noetherian

[resp.

artinian

] modules satisfy the ACC [resp. DCC] on

submodules

. Indeed the Krull-Schmidt theorem also appears in the

context

of module theory. (Sometimes, as in Lang’s

Algebra

, this result is called the Krull-Remak-Schmidt theorem.)

Krull-Schmidt theorem (for modules):

A nonzero module that is both noetherian and artinian can be expressed as the

direct sum

of finitely many

indecomposable modules

. These indecomposable summands are uniquely determined up to

isomorphism

and

permutation

.

References.

•

Hungerford, T.,

Algebra

. New York: Springer, 1974.

•

Lang, S.,

Algebra

. (3d ed.), New York: Springer, 2002.

Title

Krull-Schmidt theorem

Canonical name

KrullSchmidtTheorem

Date of creation

2013-03-22 15:24:00

Last modified on

2013-03-22 15:24:00

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

24

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 16P40

Classification

msc 16P20

Classification

msc 16D70

Classification

msc 20E34

Classification

msc 20-00

Synonym

Krull-Remak-Schmidt theorem

Related topic

IndecomposableGroup

Defines

ascending chain condition

Defines

descending chain condition

Generated on Fri Feb 9 18:42:14 2018 by

LaTeXML
noetherian

noetherian

A module

M

is

noetherian

if it

satisfies

the following

equivalent

conditions:

•

the

ascending chain condition

holds for

submodules

of

M

;

•

every nonempty family of submodules of

M

has a

maximal element

;

•

every submodule of

M

is

finitely generated

.

A ring

R

is

left noetherian

if it is noetherian as a

left module

over itself
(i.e. if

R

R

is a ),
and

right noetherian

if it is noetherian as a right module over itself
(i.e. if

R

R

is an ),
and simply

noetherian

if both conditions hold.

Title

noetherian

Canonical name

Noetherian

Date of creation

2013-03-22 12:26:53

Last modified on

2013-03-22 12:26:53

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

5

Author

antizeus (11)

Entry type

Definition

Classification

msc 16P40

Synonym

left noetherian

Synonym

right noetherian

Related topic

Artinian

Related topic

Noetherian

Related topic

HollowMatrixRings

Generated on Fri Feb 9 18:42:16 2018 by

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Goldie ring

Goldie ring

Let

R

be a ring. If the set of

annihilators

{

r

.

ann

⁡

(

x

)

∣

x

∈

R

}

satisifies the

ascending chain condition

, then

R

is said to satisfy the

ascending chain condition on right annihilators

.

A ring

R

is called a

right Goldie ring

if it satisfies the ascending chain condition on right annihilators and

R

R

is a

module of finite rank

.

Left Goldie ring

is defined similarly. If the

context

makes it clear on which side the ring operates, then such a ring is simply called a

Goldie ring

.

A

right Noetherian ring

is right Goldie.

Title

Goldie ring

Canonical name

GoldieRing

Date of creation

2013-03-22 14:04:10

Last modified on

2013-03-22 14:04:10

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 16P60

Synonym

Goldie

Related topic

UniformDimension

Defines

left Goldie

Defines

right Goldie

Defines

left Goldie ring

Defines

right Goldie ring

Generated on Fri Feb 9 18:42:19 2018 by

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uniform dimension

uniform dimension

Let

M

be a module over a ring

R

, and suppose that

M

contains no infinite

direct sums

of non-zero

submodules

. (This is the same as saying that

M

is a

module of finite rank

.)

Then there exists an integer

n

such that

M

contains an

essential submodule

N

where

N

=

U

1

⊕

U

2

⊕

…

⊕

U

n

is a direct sum of

n

uniform submodules

.

This number

n

does not depend on the choice of

N

or the decomposition into uniform submodules.

We call

n

the

uniform dimension

of

M

. Sometimes this is written

u

-

dim

⁡

M

=

n

.

If

R

is a field

K

, and

M

is a

finite-dimensional

vector space

over

K

, then

u

-

dim

⁡

M

=

dim

K

⁡

M

.

u

-

dim

⁡

M

=

0

if and only if

M

=

0

.

Title

uniform dimension

Canonical name

UniformDimension

Date of creation

2013-03-22 14:02:59

Last modified on

2013-03-22 14:02:59

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

7

Author

mclase (549)

Entry type

Definition

Classification

msc 16P60

Related topic

GoldieRing

Generated on Fri Feb 9 18:42:21 2018 by

LaTeXML
IBN

IBN

Bases of a Module

Like a

vector space

over a field, one can define a basis of a module

M

over a general ring

R

with 1. To simplify matter, suppose

R

is

commutative

with

1

and

M

is unital. A basis of

M

is a subset

B

=

{

b

i

∣

i

∈

I

}

of

M

, where

I

is some ordered

index set

, such that every

element

m

∈

M

can be uniquely written as a

linear combination

of elements from

B

:

m

=

∑

i

∈

I

r

i

⁢

b

i

such that all but a finite number of

r

i

=

0

.

As the above example shows, the commutativity of

R

is not required, and

M

can be assumed either as a left or

right module

of

R

(in the example above, we could take

M

to be the left

R

-module).

However, unlike a vector space, a module may not have a basis. If it does, it is a called a

free module

. Vector spaces are

examples of free modules

over fields or

division rings

. If a free module

M

(over

R

) has a finite basis with

cardinality

n

, we often write

R

n

as an

isomorphic

copy of

M

.

Suppose that we are given a free module

M

over

R

, and two bases

B

1

≠

B

2

for

M

, is

|

B

1

|

=

|

B

2

|

⁢

?

We know that this is true if

R

is a field or even a division ring. But in general, the equality fails. Nevertheless, it is a fact that if

B

1

is finite, so is

B

2

. So the finiteness of basis in a free module

M

over

R

is preserved when we go from one basis to another. When

M

has a finite basis, we say that

M

has

finite rank

(without saying what rank is!).

Now, even if

M

has finite rank, the cardinality of one basis may still be different from the cardinality of another. In other words,

R

m

may be isomorphic to

R

n

without

m

and

n

being equal.

Invariant Basis Number

A ring

R

is said to have

IBN

, or

invariant basis number

if whenever

R

m

≅

R

n

where

m

,

n

<

∞

,

m

=

n

. The positive integer

n

in this case is called the

rank

of module

M

. To rephrase, when

F

is a free

R

-

module of finite rank

, then

R

has IBN iff

F

has unique finite rank. Also,

R

has IBN iff all finite dimensional

invertible matrices

over

R

are

square matrices

.

Examples

1.

If

R

is commutative, then

R

has IBN.

2.

If

R

is a division ring, then

R

has IBN.

3.

An example of a ring

R

not having IBN can be found as follows: let

V

be a

countably infinite

dimensional vector space over a field. Let

R

be the

endomorphism ring

over

V

. Then

R

=

R

⊕

R

and thus

R

m

=

R

n

for any pairs of

(

m

,

n

)

.

Title

IBN

Canonical name

IBN

Date of creation

2013-03-22 14:51:45

Last modified on

2013-03-22 14:51:45

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16P99

Synonym

invariant basis number

Synonym

invariant dimension property

Related topic

ExampleOfFreeModuleWithBasesOfDiffrentCardinality

Defines

basis of a module

Defines

finite rank

Defines

rank of a module

Generated on Fri Feb 9 18:42:23 2018 by

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stably finite

stably finite

A ring

R

is

stably finite

if the

matrix rings

𝕄

n

⁢

(

R

)

are

Dedekind-finite

for all

n

.

Title

stably finite

Canonical name

StablyFinite

Date of creation

2013-03-22 14:18:26

Last modified on

2013-03-22 14:18:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16P99

Synonym

von Neumann finite

Synonym

directly finite

Generated on Fri Feb 9 18:42:25 2018 by

LaTeXML