aarjaneiro/planetmath huggingface dataset:
differential graded algebra

differential graded algebra

Let

R

be a

commutative ring

. A

differential graded algebra

(or

DG algebra

) over

R

is a complex

(

A

,

∂

A

)

of

R

-modules with an

element

1

∈

A

(the unit) and a degree zero chain map

A

⊗

R

A

→

A

that is

unitary

:

a

⁢

1

=

a

=

1

⁢

a

, and is

associative

:

a

⁢

(

b

⁢

c

)

=

(

a

⁢

b

)

⁢

c

. We also will stipulate that a DG algebra is graded

commutative

; that is for each

x

,

y

∈

A

, we have

x

⁢

y

=

(

-

1

)

|

x

|

⁢

|

y

|

⁢

y

⁢

x

where

|

x

|

means the degree of

x

. Also, we assume that

A

i

=

0

for

i

<

0

. Without these final

assumptions

, we will say that

A

is an

associative

DG algebra.

The fact that the

product

is a chain map of degree zero is best described by the Leibniz Rule; that is, for each

x

,

y

∈

A

, we have

∂

A

⁡

(

x

⁢

y

)

=

∂

A

⁡

(

x

)

⁡

y

+

(

-

1

)

|

x

|

⁢

x

⁢

∂

A

⁡

(

y

)

.

Title

differential graded algebra

Canonical name

DifferentialGradedAlgebra

Date of creation

2013-03-22 15:34:43

Last modified on

2013-03-22 15:34:43

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16E45

Synonym

DG Algebra

Generated on Fri Feb 9 18:40:43 2018 by

LaTeXML
nested ideals in von Neumann regular ring

nested ideals in von Neumann regular ring

Theorem

.

Let

𝔞

be an ideal of the

von Neumann regular ring

R

.  Then

𝔞

itself is a von Neumann regular ring and any ideal

𝔟

of

𝔞

is likewise an ideal of

R

.

Proof.

If

a

∈

𝔞

, then

a

⁢

s

⁢

a

=

a

for some

s

∈

R

.  Setting

t

=

s

⁢

a

⁢

s

we see that

t

belongs to the ideal

𝔞

and

a

⁢

t

⁢

a

=

a

⁢

(

s

⁢

a

⁢

s

)

⁢

a

=

(

a

⁢

s

⁢

a

)

⁢

s

⁢

a

=

a

⁢

s

⁢

a

=

a

.

Secondly, we have to show that whenever

b

∈

𝔟

⊆

𝔞

and

r

∈

R

, then both

b

⁢

r

and

r

⁢

b

lie in

𝔟

.  Now,

b

⁢

r

∈

𝔞

because

𝔞

is an ideal of

R

.  Thus there is an element

x

in

𝔞

satisfying

b

⁢

r

⁢

x

⁢

b

⁢

r

=

b

⁢

r

.  Since

r

⁢

x

⁢

b

⁢

r

belongs to

𝔞

and

𝔟

is assumed to be an ideal of

𝔞

, we conclude that the product

b

⋅

r

⁢

x

⁢

b

⁢

r

must lie in

𝔟

, i.e.

b

⁢

r

∈

𝔟

.  Similarly it can be shown that

r

⁢

b

∈

𝔟

.
∎

References

1

David M. Burton:

A first course in rings and ideals

.  Addison-Wesley.  Reading, Massachusetts (1970).

Title

nested ideals in von Neumann regular ring

Canonical name

NestedIdealsInVonNeumannRegularRing

Date of creation

2013-03-22 14:48:24

Last modified on

2013-03-22 14:48:24

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 16E50

Generated on Fri Feb 9 18:40:45 2018 by

LaTeXML
von Neumann regular

von Neumann regular

An

element

a

of a ring

R

is said to be

von Neumann regular

if there
exists

b

∈

R

such that

a

⁢

b

⁢

a

=

a

. Such an element

b

is known as a

of

a

.

For example, any unit in a ring is von Neumann regular. Also, any

idempotent element

is von Neumann regular. For a non-unit, non-idempotent von Nuemann

regular element

, take

M

2

⁢

(

ℝ

)

, the ring of

2

×

2

matrices over

ℝ

. Then

(

2

0

0

0

)

=

(

2

0

0

0

)

⁢

(

1

2

0

0

0

)

⁢

(

2

0

0

0

)

is von Neumann regular. In fact, we can replace

2

with any non-zero

r

∈

ℝ

and the resulting matrix is also von Neumann regular. There are several ways to generalize this example. One way is take a

central idempotent

e

in any ring

R

, and any

r

⁢

s

=

f

with

e

⁢

f

=

e

. Then

r

⁢

e

is von Neumann regular, with

s

,

s

⁢

e

and

s

⁢

f

all as pseudoinverses. In another

generalization

, we have two rings

R

,

S

where

R

is an algebra over

S

. Take any

idempotent

e

∈

R

, and any invertible element

s

∈

S

such that

s

commutes with

e

. Then

s

⁢

e

is von Neumann regular.

A ring

R

is said to be a

von Neumann regular ring

(or simply
a

regular ring

, if the is clear from

context

)
if every element of

R

is von Neumann regular.

For example, any

division ring

is von Neumann regular, and so is any ring of matrices over a division ring. In general, any

semisimple ring

is von Neumann regular.

Remark

. Note that

regular ring

in the sense of von Neumann should not be confused with

regular ring

in the sense of , which is a

Noetherian ring

whose

localization

at every

prime ideal

is a

regular local ring

.

Title

von Neumann regular

Canonical name

VonNeumannRegular

Date of creation

2013-03-22 12:56:18

Last modified on

2013-03-22 12:56:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16E50

Defines

von Neumann regular ring

Defines

regular ring

Defines

pseudoinverse

Generated on Fri Feb 9 18:40:46 2018 by

LaTeXML
Brauer group

Brauer group

1

Algebraic view

Let

K

be a field. The

Brauer group

Br

⁡

(

K

)

of

K

is the set of all

equivalence classes

of

central simple algebras

over

K

, where two central simple algebras

A

and

B

are

equivalent

if there exists a

division ring

D

over

K

and

natural numbers

n

,

m

such that

A

(resp.

B

) is

isomorphic

to the ring of

n

×

n

(resp.

m

×

m

) matrices with coefficients in

D

.

The

group operation

in

Br

⁡

(

K

)

is given by

tensor product

: for any two central simple algebras

A

,

B

over

K

, their

product

in

Br

⁡

(

K

)

is the central simple algebra

A

⊗

K

B

. The

identity element

in

Br

⁡

(

K

)

is the class of

K

itself, and the

inverse

of a central simple algebra

A

is
the

opposite

algebra

A

opp

defined by reversing
the order of the multiplication

operation

of

A

.

2

Cohomological view

The Brauer group of

K

is naturally isomorphic to the second Galois cohomology group

H

2

⁢

(

Gal

⁡

(

K

sep

/

K

)

,

(

K

sep

)

×

)

.
See

http://www.math.harvard.edu/ elkies/M250.01/index.html

http://www.math.harvard.edu/ elkies/M250.01/index.html

Theorem

12 and succeeding remarks.

Title

Brauer group

Canonical name

BrauerGroup

Date of creation

2013-03-22 11:49:31

Last modified on

2013-03-22 11:49:31

Owner

djao (24)

Last modified by

djao (24)

Numerical id

13

Author

djao (24)

Entry type

Definition

Classification

msc 16K50

Defines

opposite algebra

Generated on Fri Feb 9 18:40:51 2018 by

LaTeXML
division ring

division ring

A

division ring

is a ring

D

with

identity

such that

•

1

≠

0

•

For all nonzero

a

∈

D

, there exists

b

∈

D

with

a

⋅

b

=

b

⋅

a

=

1

Every field is a

commutative

division ring. The

Hamiltonian quaternions

are an example of a division ring which is not a field.

Title

division ring

Canonical

name

DivisionRing

Date of creation

2013-03-22 11:48:46

Last modified on

2013-03-22 11:48:46

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 16K99

Classification

msc 81P05

Synonym

skew field

Generated on Fri Feb 9 18:40:55 2018 by

LaTeXML
semi-local ring

semi-local ring

A ring

R

is

semi-local

if

R

/

r

⁢

a

⁢

d

⁢

(

R

)

is an Artinian ring, where

r

⁢

a

⁢

d

⁢

(

R

)

denotes the

Jacobson radical

of

R

. In the case that

R

is

commutative

, this reduces to the definition that

R

is

semi-local

if

R

has finitely many

maximal ideals

. Note that

finite rings

are trivially semi-local.

Title

semi-local ring

Canonical name

SemilocalRing

Date of creation

2013-03-22 12:56:42

Last modified on

2013-03-22 12:56:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16L30

Classification

msc 13H99

Synonym

semilocal ring

Related topic

LocalRing

Generated on Fri Feb 9 18:40:57 2018 by

LaTeXML
a ring modulo its Jacobson radical is semiprimitive

a ring modulo its Jacobson radical is semiprimitive

Let

R

be a ring. Then

J

⁢

(

R

/

J

⁢

(

R

)

)

=

(

0

)

.

Proof:

We will only prove this in the case where

R

is a

unital ring

(although it is true without this

assumption

).

Let

[

u

]

∈

J

⁢

(

R

/

J

⁢

(

R

)

)

.
By one of the

characterizations

of the

Jacobson radical

,

1

-

[

r

]

⁢

[

u

]

is

left invertible

for all

r

∈

R

,
so there exists

v

∈

R

such that

[

v

]

⁢

(

1

-

[

r

]

⁢

[

u

]

)

=

1

.

Then

v

⁢

(

1

-

r

⁢

u

)

=

1

-

a

for some

a

∈

J

⁢

(

R

)

.
There is a

w

∈

R

such that

w

⁢

(

1

-

a

)

=

1

,
and we have

w

⁢

v

⁢

(

1

-

r

⁢

u

)

=

1

.

Since this holds for all

r

∈

R

,
it follows that

u

∈

J

⁢

(

R

)

, and therefore

[

u

]

=

0

.

Title

a

ring modulo its Jacobson radical is semiprimitive

Canonical

name

ARingModuloItsJacobsonRadicalIsSemiprimitive

Date of creation

2013-03-22 12:49:34

Last modified on

2013-03-22 12:49:34

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Theorem

Classification

msc 16N20

Generated on Fri Feb 9 18:41:00 2018 by

LaTeXML
examples of semiprimitive rings

examples of semiprimitive rings

Examples of semiprimitive rings

:

The integers

Z

:

Since

ℤ

is

commutative

, any

left ideal

is two-sided. So the

maximal

left ideals of

ℤ

are the

maximal ideals

of

ℤ

, which are the ideals

p

⁢

ℤ

for

p

prime.
So

J

⁢

(

ℤ

)

=

⋂

p

p

⁢

ℤ

=

(

0

)

,
as there are infinitely many primes.

A

matrix ring

M

n

⁢

(

D

)

over a

division ring

D

:

The ring

M

n

⁢

(

D

)

is simple, so the only

proper ideal

is

(

0

)

. Thus

J

⁢

(

M

n

⁢

(

D

)

)

=

(

0

)

.

A

polynomial ring

R

⁢

[

x

]

over an

integral domain

R

:

Take

a

∈

J

⁢

(

R

⁢

[

x

]

)

with

a

≠

0

.
Then

a

⁢

x

∈

J

⁢

(

R

⁢

[

x

]

)

, since

J

⁢

(

R

⁢

[

x

]

)

is an ideal, and

deg

⁡

(

a

⁢

x

)

≥

1

.
By one of the alternate

characterizations

of the

Jacobson radical

,

1

-

a

⁢

x

is a unit.
But

deg

⁡

(

1

-

a

⁢

x

)

=

max

⁡

{

deg

⁡

(

1

)

,

deg

⁡

(

a

⁢

x

)

}

≥

1

.
So

1

-

a

⁢

x

is not a unit, and by this

contradiction

we see that

J

⁢

(

R

⁢

[

x

]

)

=

(

0

)

.

Title

examples of semiprimitive rings

Canonical name

ExamplesOfSemiprimitiveRings

Date of creation

2013-03-22 12:50:39

Last modified on

2013-03-22 12:50:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Example

Classification

msc 16N20

Generated on Fri Feb 9 18:41:02 2018 by

LaTeXML
Jacobson radical

Jacobson radical

The

Jacobson radical

J

⁢

(

R

)

of a

unital ring

R

is the

intersection

of the

annihilators

of simple (

http://planetmath.org/SimpleModule

) left

R

-modules.

The following are alternative

characterizations

of the Jacobson radical

J

⁢

(

R

)

:

1.

The intersection of all left

primitive ideals

.

2.

The intersection of all

maximal

left ideals

.

3.

The set of all

t

∈

R

such that for all

r

∈

R

,

1

-

r

⁢

t

is

left invertible

(i.e. there exists

u

such that

u

⁢

(

1

-

r

⁢

t

)

=

1

).

4.

The largest ideal

I

such that for all

v

∈

I

,

1

-

v

is a
unit in

R

.

5.

(1) - (3) with “left” replaced by “right” and

r

⁢

t

replaced by

t

⁢

r

.

If

R

is

commutative

and

finitely generated

, then

J

⁢

(

R

)

=

{

x

∈

R

∣

x

n

=

0

⁢

for some

⁢

n

∈

ℕ

}

=

Nil

⁡

(

R

)

.

The Jacobson radical can also be defined for non-unital rings.
To do this, we first define a

binary operation

∘

on the ring

R

by

x

∘

y

=

x

+

y

-

x

⁢

y

for all

x

,

y

∈

R

.
Then

(

R

,

∘

)

is a monoid,
and the Jacobson radical is defined to be the largest ideal

I

of

R

such that

(

I

,

∘

)

is a group.
If

R

is unital, this is

equivalent

to the definitions given earlier.

Title

Jacobson radical

Canonical name

JacobsonRadical

Date of creation

2013-03-22 12:36:11

Last modified on

2013-03-22 12:36:11

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

19

Author

yark (2760)

Entry type

Definition

Classification

msc 16N20

Related topic

Annihilator

Related topic

RadicalOfAnIdeal

Related topic

SimpleModule

Related topic

Nilradical

Related topic

RadicalTheory

Related topic

QuasiRegularity

Generated on Fri Feb 9 18:41:04 2018 by

LaTeXML
properties of the Jacobson radical

properties of the Jacobson radical

Theorem

:

Let

R

,

T

be rings and

φ

:

R

→

T

be a

surjective

homomorphism

. Then

φ

⁢

(

J

⁢

(

R

)

)

⊆

J

⁢

(

T

)

.

Proof:

We shall use the

characterization

of the

Jacobson radical

as the set of all

a

∈

R

such that for all

r

∈

R

,

1

-

r

⁢

a

is

left invertible

.

Let

a

∈

J

⁢

(

R

)

,

t

∈

T

. We claim that

1

-

t

⁢

φ

⁢

(

a

)

is left invertible:

Since

φ

is surjective,

t

=

φ

⁢

(

r

)

for some

r

∈

R

. Since

a

∈

J

⁢

(

R

)

, we know

1

-

r

⁢

a

is left invertible, so there exists

u

∈

R

such that

u

⁢

(

1

-

r

⁢

a

)

=

1

. Then we have

φ

⁢

(

u

)

⁢

(

φ

⁢

(

1

)

-

φ

⁢

(

r

)

⁢

φ

⁢

(

a

)

)

=

φ

⁢

(

u

)

⁢

φ

⁢

(

1

-

r

⁢

a

)

=

φ

⁢

(

1

)

=

1

So

φ

⁢

(

a

)

∈

J

⁢

(

T

)

as required.

Theorem:

Let

R

,

T

be rings. Then

J

⁢

(

R

×

T

)

⊆

J

⁢

(

R

)

×

J

⁢

(

T

)

.

Proof:

Let

π

1

:

R

×

T

→

R

be a (surjective)

projection

.
By the previous theorem,

π

1

⁢

(

J

⁢

(

R

×

T

)

)

⊆

J

⁢

(

R

)

.

Similarly let

π

2

:

R

×

T

→

T

be a (surjective) projection. We see that

π

2

⁢

(

J

⁢

(

R

×

T

)

)

⊆

J

⁢

(

T

)

.

Now take

(

a

,

b

)

∈

J

⁢

(

R

×

T

)

. Note that

a

=

π

1

⁢

(

a

,

b

)

∈

J

⁢

(

R

)

and

b

=

π

2

⁢

(

a

,

b

)

∈

J

⁢

(

T

)

. Hence

(

a

,

b

)

∈

J

⁢

(

R

)

×

J

⁢

(

T

)

as required.

Title

properties of the Jacobson radical

Canonical name

PropertiesOfTheJacobsonRadical

Date of creation

2013-03-22 12:49:43

Last modified on

2013-03-22 12:49:43

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Result

Classification

msc 16N20

Generated on Fri Feb 9 18:41:13 2018 by

LaTeXML
quasi-regularity

quasi-regularity

An

element

x

of a ring is called

right quasi-regular

[resp.

left quasi-regular

] if there
is an element

y

in the ring such that

x

+

y

+

x

⁢

y

=

0

[resp.

x

+

y

+

y

⁢

x

=

0

].

For calculations with quasi-regularity, it is useful to introduce the

operation

*

defined:

x

*

y

=

x

+

y

+

x

⁢

y

.

Thus

x

is right quasi-regular if there is an element

y

such that

x

*

y

=

0

.
The operation

*

is easily demonstrated to be

associative

, and

x

*

0

=

0

*

x

=

x

for all

x

.

An element

x

is called

quasi-regular

if it is both left and right quasi-regular.
In this case, there are elements

y

and

z

such that

x

+

y

+

x

⁢

y

=

0

=

x

+

z

+

z

⁢

x

(equivalently,

x

*

y

=

z

*

x

=

0

).
A calculation shows that

y

=

0

*

y

=

(

z

*

x

)

*

y

=

z

*

(

x

*

y

)

=

z

.

So

y

=

z

is a unique element, depending on

x

, called the

quasi-inverse

of

x

.

An ideal (one- or two-sided) of a ring is called

quasi-regular

if each of its elements is quasi-regular. Similarly, a ring is called

quasi-regular

if each of its elements is quasi-regular (such rings cannot have an

identity element

).

Lemma

.

Let

A

be an ideal (one- or two-sided) in a ring

R

.
If each element of

A

is right quasi-regular, then

A

is a quasi-regular ideal.

This lemma means that there is no extra generality gained in defining terms such as right quasi-regular

left ideal

, etc.

Quasi-regularity is important because it provides

elementary

characterizations

of the

Jacobson radical

for rings without an identity element:

•

The Jacobson radical of a ring is the sum of all quasi-regular left (or right) ideals.

•

The Jacobson radical of a ring is the largest quasi-regular ideal of the ring.

For rings with an identity element, note that

x

is [right, left] quasi-regular if and only if

1

+

x

is [right, left]

invertible

in the ring.

Title

quasi-regularity

Canonical

name

Quasiregularity

Date of creation

2013-03-22 13:12:59

Last modified on

2013-03-22 13:12:59

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Definition

Classification

msc 16N20

Synonym

quasi regular

Synonym

quasi regularity

Related topic

JacobsonRadical

Related topic

RegularIdeal

Related topic

HomotopesAndIsotopesOfAlgebras

Defines

quasi-regular

Defines

right quasi-regular

Defines

left quasi-regular

Defines

quasi-inverse

Defines

quasi-regular ideal

Defines

quasi-regular ring

Generated on Fri Feb 9 18:41:15 2018 by

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semiprimitive ring

semiprimitive ring

A ring is said to be

semiprimitive

if its

Jacobson radical

is the

zero ideal

.

Any

simple ring

is automatically semiprimitive.

A finite

direct product

of

matrix rings

over

division rings

can be shown to be semiprimitive and both left and right

Artinian

.

The

Artin-Wedderburn Theorem

(

http://planetmath.org/WedderburnArtinTheorem

) states that any semiprimitive ring which is left or right Artinian is

isomorphic

to a finite direct product of matrix rings over division rings.

Note

:
The semiprimitive condition is sometimes also referred to as a

semisimple

,

Jacobson semisimple

, or

J-semisimple

. Furthermore, when either of the last two names are used, the adjective ’semisimple’ is frequently intended to refer to a ring that is semiprimitive and Artinian (see the entry on

semisimple rings

(

http://planetmath.org/SemisimpleRing2

)).

Title

semiprimitive ring

Canonical name

SemiprimitiveRing

Date of creation

2013-03-22 12:36:14

Last modified on

2013-03-22 12:36:14

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Definition

Classification

msc 16N20

Synonym

semisimple ring

Synonym

Jacobson semisimple ring

Synonym

J-semisimple ring

Synonym

semi-primitive ring

Synonym

semi-simple ring

Synonym

Jacobson semi-simple ring

Synonym

J-semi-simple ring

Related topic

SemisimpleRing2

Related topic

WedderburnArtinTheorem

Defines

semiprimitivity

Defines

semiprimitive

Defines

semisimple

Defines

Jacobson semisimple

Defines

J-semisimple

Defines

semi-primitivity

Defines

semi-primitive

Defines

semi-simple

Defines

Jacobson semi-simple

Defines

J-semi-simple

Generated on Fri Feb 9 18:41:16 2018 by

LaTeXML
a characterization of the radical of an ideal

a characterization of the radical of an ideal

Proposition 1

.

Let

I

be an ideal in a ring

R

, and

I

be its

radical

. Then

I

is the

intersection

of all

prime ideals

containing

I

.

Proof.

Suppose

x

∈

I

, and

P

is a prime ideal containing

I

. Then

R

-

P

is an

m

-system (

http://planetmath.org/MSystem

). If

x

∈

R

-

P

, then

(

R

-

P

)

∩

I

≠

∅

, contradicting the

assumption

that

I

⊆

P

. Therefore

x

∉

R

-

P

. In other words,

x

∈

P

, and we have one of the

inclusions

.

Conversely

, suppose

x

∉

I

. Then there is an

m

-system

S

containing

x

such that

S

∩

I

=

∅

. Enlarge

I

to a prime ideal

P

disjoint

from

S

, so that

x

∉

P

(we can do this; for a proof, see the second remark in this entry (

http://planetmath.org/MSystem

)). By contrapositivity, we have the other inclusion.
∎

Remark

. This shows that every prime ideal is a radical ideal: for

P

is the intersection of all prime ideals containing

P

, and if

P

is itself prime, then

P

=

P

.

Title

a

characterization

of the radical of an ideal

Canonical name

ACharacterizationOfTheRadicalOfAnIdeal

Date of creation

2013-03-22 18:04:50

Last modified on

2013-03-22 18:04:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 16N40

Classification

msc 13-00

Classification

msc 14A05

Generated on Fri Feb 9 18:41:25 2018 by

LaTeXML
examples of radicals of ideals in commutative rings

examples of radicals of ideals in commutative rings

Let

R

be a

commutative ring

. Recall, that ideals

I

,

J

in

R

are called

coprime

iff

I

+

J

=

R

. It can be shown, that if

I

,

J

are coprime, then

I

⁢

J

=

I

∩

J

. Elements

x

1

,

…

,

x

n

∈

R

are called

pairwise coprime

iff

(

x

i

)

+

(

x

j

)

=

R

for

i

≠

j

. It follows by induction, that for pairwise coprime

x

1

,

…

,

x

n

∈

R

we have

(

x

1

⁢

⋯

⁢

x

n

)

=

(

x

1

)

∩

⋯

∩

(

x

n

)

,

Let

x

∈

R

be such that

x

=

p

1

α

1

⁢

⋯

⁢

p

n

α

n

,

for some

prime elements

p

i

∈

R

,

α

i

∈

ℕ

and assume that

p

1

,

…

,

p

n

are coprime. Denote by

x

¯

=

p

1

⁢

⋯

⁢

p

n

.

We shall denote by

r

⁢

(

I

)

the

radical

of an ideal

I

⊆

R

.

Proposition.

r

⁢

(

(

x

)

)

=

(

x

¯

)

.

Proof.

,,

⊇

” Let

α

=

max

⁢

(

α

1

,

…

,

α

n

)

. Then we have

x

¯

α

=

(

p

1

⁢

⋯

⁢

p

n

)

α

=

p

1

α

⁢

⋯

⁢

p

n

α

=

p

1

α

-

α

1

⁢

⋯

⁢

p

n

α

-

α

n

⁢

p

1

α

1

⁢

⋯

⁢

p

n

α

n

=

y

⁢

x

and thus

x

¯

α

∈

(

x

)

. This shows the first inclusion.

,,

⊆

” Assume that

y

∈

r

⁢

(

(

x

)

)

and

y

≠

0

. Then there is

n

∈

ℕ

such that

y

n

∈

(

x

)

. Thus

x

divides

y

n

. Of course for any

i

∈

{

1

,

…

,

n

}

we have that

p

i

divides

x

. Thus

p

i

divides

y

n

and since

p

i

is prime, we obtain that

p

i

divides

y

. Now for

i

≠

j

elements

p

i

and

p

j

are coprime, thus

x

¯

divides

y

and therefore

y

∈

(

x

¯

)

, which

completes

the proof.

□

Remark.

If we assume that

R

is a PID (and thus UFD), then the previous proposition gives us the full

characterization of radicals of ideals

in

R

. In particular an ideal in PID is radical if and only if it is generated by an element of the form

p

1

⁢

⋯

⁢

p

n

, where for

i

≠

j

elements

p

i

and

p

j

are not associated primes.

Examples.

Consider

ring of integers

ℤ

. Then we have:

r

⁢

(

(

12

)

)

=

(

6

)

;

r

⁢

(

(

9

)

)

=

(

3

)

;

r

⁢

(

(

7

)

)

=

(

7

)

;

r

⁢

(

(

1125

)

)

=

(

15

)

.

Title

examples of radicals of ideals in commutative rings

Canonical name

ExamplesOfRadicalsOfIdealsInCommutativeRings

Date of creation

2013-03-22 19:04:34

Last modified on

2013-03-22 19:04:34

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Example

Classification

msc 16N40

Classification

msc 14A05

Classification

msc 13-00

Generated on Fri Feb 9 18:41:28 2018 by

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Koethe conjecture

Koethe conjecture

The

Koethe Conjecture

is the statement that for any pair of

nil right ideals

A

and

B

in any ring

R

, the sum

A

+

B

is also nil.

If either of

A

or

B

is a

two-sided ideal

, it is easy to see that

A

+

B

is nil. (See

properties of nil and nilpotent ideals

.)

In particular, this means that the Koethe conjecture is true for

commutative rings

.

It has been shown to be true for many classes of rings, but the general statement is still unproven, and no counter example has been found.

Title

Koethe conjecture

Canonical name

KoetheConjecture

Date of creation

2013-03-22 13:13:27

Last modified on

2013-03-22 13:13:27

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

5

Author

mclase (549)

Entry type

Conjecture

Classification

msc 16N40

Related topic

NilAndNilpotentIdeals

Related topic

PropertiesOfNilAndNilpotentIdeals

Generated on Fri Feb 9 18:41:30 2018 by

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