aarjaneiro/planetmath huggingface dataset:
example of reducible and irreducible G-modules

example of reducible and irreducible

G

-modules

Let

G

=

S

r

, the

permutation group

on

r

elements, and

N

=

k

r

where

k

is an arbitrary field. Consider the

permutation representation

of

G

on

N

given by

σ

⁢

(

a

1

,

…

,

a

r

)

=

(

a

σ

⁢

(

1

)

,

…

,

a

σ

⁢

(

r

)

)

,

σ

∈

S

r

,

a

i

∈

k

If

r

>

1

, we can define two

submodules

of

N

, called the

trace

and

augmentation

, as

N

′

=

{

(

a

,

a

,

…

,

a

)

}

N

′′

=

{

(

a

1

,

a

2

,

…

,

a

r

)

|

∑

a

i

=

0

.

}

Clearly both

N

′

and

N

′′

are stable under the action of

G

and thus in fact form submodules of

N

.

If the

characteristic

of

k

divides

r

, then obviously

N

′′

⊃

N

′

. Otherwise,

N

′′

is a simple (

irreducible

)

G

-module. For suppose

N

′′

has a nontrivial submodule

M

, and choose a nonzero

u

∈

M

. Then some pair of coordinates of

u

are unequal, for if not, then

u

=

(

a

,

…

,

a

)

and then

u

∉

N

′′

because of the

restriction

on the characteristic of

k

forces

r

⁢

a

≠

0

. So apply a suitable element of

G

to get another element of

M

,

v

=

(

b

1

,

b

2

,

…

,

b

r

)

where

b

1

≠

b

2

(note here that we use the fact that

M

is a submodule and thus is stable under the action of

G

).

But now

(

12

)

⁢

v

-

e

⁢

v

=

(

b

1

-

b

2

,

b

2

-

b

1

,

0

,

…

,

0

)

is also in

M

, so

w

=

(

1

,

-

1

,

0

,

…

,

0

)

∈

M

. It is obvious that by multiplying

w

by elements of

k

and by permuting, we can obtain any element of

N

′′

and thus

M

=

N

′′

. Thus

N

′′

is simple.

It is also obvious that

N

=

N

′

⊕

N

′′

.

Title

example of reducible and irreducible

G

-modules

Canonical name

ExampleOfReducibleAndIrreducibleGmodules

Date of creation

2013-03-22 16:37:50

Last modified on

2013-03-22 16:37:50

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Example

Classification

msc 16D60

Defines

augmentation

Generated on Fri Feb 9 18:39:12 2018 by

LaTeXML
properties of semisimple modules

properties of semisimple modules

Let

R

be a ring. Recall that

R

-module

M

is called

semisimple

iff

M

is a

direct sum

of

simple module

.

Proposition

.

The following are

equivalent

for

R

-module

M

:

1.

M

is semisimple;

2.

M

is generated by its simple

submodules

;

3.

for every submodule

N

⊆

M

there exists a submodule

N

′

⊆

M

such that

M

=

N

⊕

N

′

.

Title

properties of semisimple modules

Canonical name

PropertiesOfSemisimpleModules

Date of creation

2013-03-22 18:53:27

Last modified on

2013-03-22 18:53:27

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 16D60

Generated on Fri Feb 9 18:39:14 2018 by

LaTeXML
semisimple ring

semisimple ring

A ring

R

is (left)

semisimple

if it one of the following statements:

1.

All left

R

-modules are semisimple.

2.

All finitely-generated (

http://planetmath.org/FinitelyGeneratedRModule

) left

R

-modules are semisimple.

3.

All cyclic left

R

-modules are semisimple.

4.

The left

regular

R

-module

R

R

is semisimple.

5.

All

short exact sequences

of left

R

-modules split (

http://planetmath.org/SplitShortExactSequence

).

The last condition offers another homological characterization of a

semisimple

ring:

•

A ring

R

is (left) semisimple iff all of its

left modules

are projective (

http://planetmath.org/ProjectiveModule

).

A more ring-theorectic characterization of a (left)

semisimple ring

is:

•

A ring is left semisimple iff it is semiprimitive and

left artinian

.

In some literature, a (left) semisimple ring is defined to be a ring that is semiprimitive without necessarily being (left) artinian. Such a ring (semiprimitive) is called Jacobson semisimple, or J-semisimple, to remind us of the fact that its

Jacobson radical

is (0).

Relating to

von Neumann regular rings

, one has:

•

A ring is left semisimple iff it is von Neumann regular and

left noetherian

.

The famous

Wedderburn-Artin Theorem

that a (left) semisimple ring is

isomorphic

to a finite

direct product

of

matrix rings

over

division rings

.

The theorem implies that a left semisimplicity is synonymous with right semisimplicity, so that it is safe to drop the word left or right when referring to semisimple rings.

Title

semisimple ring

Canonical name

SemisimpleRing

Date of creation

2013-03-22 14:19:05

Last modified on

2013-03-22 14:19:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D60

Related topic

SemiprimitiveRing

Defines

semisimple

Generated on Fri Feb 9 18:39:16 2018 by

LaTeXML
simple ring

simple ring

A nonzero ring

R

is said to be a

simple ring

if it has no (two-sided) ideal other then the

zero ideal

and

R

itself.

This is

equivalent

to saying that the zero ideal is a

maximal ideal

.

If

R

is a

commutative ring

with unit, then this is equivalent to being a field.

Title

simple ring

Canonical name

SimpleRing

Date of creation

2013-03-22 11:51:07

Last modified on

2013-03-22 11:51:07

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

9

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D60

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

LaTeXML
subdirect product of rings

subdirect product of rings

A ring

R

is said to be (represented as) a

subdirect product

of a family of rings

{

R

i

:

i

∈

I

}

if:

1.

there is a

monomorphism

ε

:

R

⟶

∏

R

i

, and

2.

given 1.,

π

i

∘

ε

:

R

⟶

R

i

is

surjective

for each

i

∈

I

, where

π

i

:

∏

R

i

⟶

R

i

is the

canonical projection

map.

A

subdirect product

() of

R

is said to be

trivial

if one of the

π

i

∘

ε

:

R

⟶

R

i

is an

isomorphism

.

Direct products

and

direct sums

of rings are all examples of

subdirect products of rings

.

ℤ

does not have non-trivial direct product nor non-trivial direct sum of rings. However,

ℤ

can be represented as a non-trivial subdirect product of

ℤ

/

(

p

i

n

i

)

.

As an application of subdirect products, it can be shown that any ring can be represented as a subdirect product of

subdirectly irreducible rings

. Since a subdirectly

commutative

reduced ring

is a field, a

Boolean ring

B

can be represented as a subdirect product of

ℤ

2

. Furthermore, if this Boolean ring

B

is finite, the subdirect product becomes a direct product . Consequently,

B

has

2

n

elements

, where

n

is the number of copies of

ℤ

2

.

Title

subdirect product of rings

Canonical name

SubdirectProductOfRings

Date of creation

2013-03-22 14:19:11

Last modified on

2013-03-22 14:19:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D70

Classification

msc 16S60

Synonym

subdirect sum

Defines

trivial subdirect product

Generated on Fri Feb 9 18:39:20 2018 by

LaTeXML
subdirectly irreducible ring

subdirectly irreducible ring

A ring

R

is said to be

subdirectly irreducible

if every

subdirect product

of

R

is

trivial

.

Equivalently, a ring

R

is

subdirectly irreducible

iff the

intersection

of

all

non-zero ideals of

R

is non-zero.

Proof.

Let

{

I

i

}

be the set of all non-zero ideals of

R

.

(

⇒

)

. Suppose first that

R

is subdirectly irreducible. If

⋂

I

i

=

0

, then

R

is a subdirect product of

R

i

:=

R

/

I

i

, for

ϵ

:

R

→

∏

R

i

given by

ϵ

⁢

(

r

)

⁢

(

i

)

=

r

+

I

i

is

injective

. If

ϵ

⁢

(

r

)

=

0

, then

r

∈

I

i

for all

i

, or

r

∈

⋂

I

i

=

0

, or

r

=

0

. But then

R

→

∏

R

i

→

R

i

given by

r

↦

r

+

I

i

is not an

isomorphism

for any

i

, contradicting the fact that

R

is subdirectly irreducible. Therefore,

⋂

I

i

≠

0

.

(

⇐

)

. Suppose next that

⋂

I

i

≠

0

. Let

R

be a subdirect product of some

R

i

, and let

J

i

:=

ker

⁡

(

R

→

∏

R

i

→

R

i

)

. Each

J

i

is an ideal of

R

. Let

J

=

⋂

J

i

. If

R

→

∏

R

i

→

R

i

is not an isomorphism (therefore not injective),

J

i

is non-zero. This means that if

R

is not subdirectly irreducible,

J

≠

0

. But

J

⊆

ker

⁡

(

R

→

∏

R

i

)

, contradicting the subdirect irreducibility of

R

. As a result, some

J

i

=

0

, or

R

→

∏

R

i

→

R

i

is an isomorphism.
∎

As an application of the above equivalence, we have that a

simple ring

is subdirectly irreducible. In

addition

, a

commutative

subdirectly irreducible

reduced ring

is a field. To see this, let

{

I

i

}

be the set of all non-zero ideals of a commutative subdirectly irreducible reduced ring

R

, and let

I

=

⋂

I

i

. So

I

≠

0

by subdirect irreducibility. Pick

0

≠

s

∈

I

. Then

s

2

⁢

R

⊆

s

⁢

R

⊆

I

. So

s

2

⁢

R

=

s

⁢

R

since

I

is

minimal

. This means

s

=

s

2

⁢

t

, or

1

=

s

⁢

t

∈

s

⁢

R

=

I

, which means

I

=

R

. Now, let any

0

≠

r

∈

R

, then

R

=

I

⊆

r

⁢

R

, so

1

=

p

⁢

r

for some

p

∈

R

, which means

R

is a field.

Title

subdirectly irreducible ring

Canonical name

SubdirectlyIrreducibleRing

Date of creation

2013-03-22 14:19:13

Last modified on

2013-03-22 14:19:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D70

Generated on Fri Feb 9 18:39:22 2018 by

LaTeXML
Wedderburn-Artin theorem

Wedderburn-Artin theorem

If

R

is a left

semisimple ring

, then

R

≅

𝕄

n

1

⁢

(

D

1

)

×

⋯

×

𝕄

n

r

⁢

(

D

r

)

where each

D

i

is a

division ring

and

𝕄

n

i

⁢

(

D

i

)

is the

matrix ring

over

D

i

,

i

=

1

,

2

,

…

,

r

. The positive integer

r

is unique, and so are the division rings (up to

permutation

).

Some immediate

consequences

of this

theorem

:

•

A simple (

http://planetmath.org/SimpleRing

)

Artinian ring

is

isomorphic

to a matrix ring over a division ring.

•

A

commutative

semisimple ring is a finite

direct product

of fields.

This theorem is a special case of the more general theorem on semiprimitive rings.

Title

Wedderburn-Artin theorem

Canonical name

WedderburnArtinTheorem

Date of creation

2013-03-22 14:19:08

Last modified on

2013-03-22 14:19:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 16D70

Synonym

structure theorem on semisimple rings

Synonym

Artin-Wedderburn theorem

Related topic

SemiprimitiveRing

Generated on Fri Feb 9 18:39:23 2018 by

LaTeXML
essential submodule

essential submodule

Let

X

be a

submodule

of a module

Y

.
We say that

X

is an

essential submodule

of

Y

,
and that

Y

is an

essential extension

of

X

,
if whenever

A

is a nonzero submodule of

Y

,
then

A

∩

X

is also nonzero.

A

monomorphism

f

:

X

→

Y

is an

essential monomorphism

if the image

im

⁢

f

is an essential submodule of

Y

.

Title

essential submodule

Canonical name

EssentialSubmodule

Date of creation

2013-03-22 12:10:00

Last modified on

2013-03-22 12:10:00

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Synonym

essential extension

Synonym

essential monomorphism

Related topic

EssentialSubgroup

Related topic

EssentialMonomorphisms

Related topic

CompleteRingOfQuotientsOfReducedCommutativeRings

Related topic

EpimorphicHull

Generated on Fri Feb 9 18:39:25 2018 by

LaTeXML
faithful module

faithful module

Let

R

be a ring, and let

M

be an

R

-module.
We say that

M

is a

faithful

R

-module
if its

annihilator

ann

R

⁢

(

M

)

is the

zero ideal

.

We say that

M

is a

fully faithful

R

-module
if every nonzero

R

-

submodule

of

M

is faithful.

Title

faithful module

Canonical name

FaithfulModule

Date of creation

2013-03-22 12:01:35

Last modified on

2013-03-22 12:01:35

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

7

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Synonym

fully faithful module

Generated on Fri Feb 9 18:39:27 2018 by

LaTeXML
hereditary ring

hereditary ring

Let

R

be a ring. A right (left)

R

-module

M

is called right (left)

hereditary

if every

submodule

of

M

is projective over

R

.

Remarks

.

•

If

M

is

semisimple

, then

M

is hereditary.

•

Suppose

M

is an

external direct sum

of hereditary right (left)

R

-modules, then

M

is itself hereditary.

A ring

R

is said to be a right (left)

hereditary ring

if all of its right (left) ideals are projective as modules over

R

. If

R

is both left and right hereditary, then

R

is simply called a hereditary ring.

Remarks.

•

Even though the notions of left and right heredity in rings are symmetrical, one does not imply the other.

•

If

R

is semisimple, then

R

is hereditary.

•

If

R

is hereditary, then every free

R

-module is a hereditary module.

•

A hereditary

integral domain

is a

Dedekind domain

, and conversely.

•

The

global dimension

of a non-semisimple hereditary ring is 1.

Title

hereditary ring

Canonical name

HereditaryRing

Date of creation

2013-03-22 14:48:50

Last modified on

2013-03-22 14:48:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D80

Classification

msc 16E60

Defines

hereditary module

Generated on Fri Feb 9 18:39:29 2018 by

LaTeXML
minimal prime ideal

minimal prime ideal

A

prime ideal

P

of a ring

R

is called a

minimal prime ideal

if it does not properly contain any other prime ideal of

R

.

If

R

is a

prime ring

, then the

zero ideal

is a prime ideal, and is thus the unique minimal prime ideal of

R

.

Title

minimal prime ideal

Canonical name

MinimalPrimeIdeal

Date of creation

2013-03-22 12:01:21

Last modified on

2013-03-22 12:01:21

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Related topic

ZeroIdeal

Generated on Fri Feb 9 18:39:31 2018 by

LaTeXML
module of finite rank

module of finite rank

Let

M

be a module,
and let

E

⁢

(

M

)

be the

injective hull

of

M

.
Then we say that

M

has
if

E

⁢

(

M

)

is a finite

direct sum

of

indecomposable

submodules

.

This turns out to be

equivalent

to the property
that

M

has no

infinite

direct sums of nonzero submodules.

Title

module of finite rank

Canonical name

ModuleOfFiniteRank

Date of creation

2013-03-22 12:03:24

Last modified on

2013-03-22 12:03:24

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Generated on Fri Feb 9 18:39:33 2018 by

LaTeXML
semihereditary ring

semihereditary ring

Let

R

be a ring. A right (left)

R

-module

M

is called right (left)

semihereditary

if every

finitely generated

submodule

of

M

is projective over

R

.

A ring

R

is said to be a right (left)

semihereditary ring

if all of its finitely generated right (left) ideals are projective as modules over

R

. If

R

is both left and right semihereditary, then

R

is simply called a semihereditary ring.

Remarks.

•

A

hereditary ring

is clearly semihereditary.

•

A ring that is left (right) semiheridtary is not necessarily right (left) semihereditary.

•

If

R

is

hereditary

, then every finitely generated submodule of a free

R

-modules is a

projective module

.

•

A semihereditary

integral domain

is a Prüfer domain, and conversely.

Title

semihereditary ring

Canonical name

SemihereditaryRing

Date of creation

2013-03-22 14:48:55

Last modified on

2013-03-22 14:48:55

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D80

Classification

msc 16E60

Defines

semihereditary module

Generated on Fri Feb 9 18:39:35 2018 by

LaTeXML
simple module

simple module

Let

R

be a ring, and let

M

be an

R

-module.
We say that

M

is a
or odule
if it contains no

submodules

other than itself and the

zero module

.

Typically, by convention, the zero module is not itself considered simple.

Title

simple module

Canonical

name

SimpleModule

Date of creation

2013-03-22 12:01:40

Last modified on

2013-03-22 12:01:40

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 16D80

Synonym

irreducible module

Related topic

JacobsonRadical

Generated on Fri Feb 9 18:39:36 2018 by

LaTeXML
superfluous submodule

superfluous submodule

Let

X

be a

submodule

of a module

Y

.
We say that

X

is a

superfluous submodule

of

Y

if whenever

A

is a submodule of

Y

such that

A

+

X

=

Y

,
then

A

=

Y

.

Title

superfluous submodule

Canonical

name

SuperfluousSubmodule

Date of creation

2013-03-22 12:10:02

Last modified on

2013-03-22 12:10:02

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Generated on Fri Feb 9 18:39:38 2018 by

LaTeXML
uniform module

uniform module

A module

M

is said to be

uniform

if any two nonzero

submodules

of

M

must have a nonzero

intersection

. This is

equivalent

to saying that any nonzero submodule is an

essential submodule

.

Title

uniform module

Canonical name

UniformModule

Date of creation

2013-03-22 11:51:20

Last modified on

2013-03-22 11:51:20

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

8

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D80

Defines

uniform submodule

Generated on Fri Feb 9 18:39:40 2018 by

LaTeXML
reflexive module

reflexive module

Let

R

be a ring, and

M

a right

R

-module. Then its dual,

M

*

, is given by

hom

⁡

(

M

,

R

)

, and has the

structure

of a

left module

over

R

. The dual of that,

M

*

*

, is in turn a right

R

-module. Fix any

m

∈

M

. Then for any

f

∈

M

*

, the

mapping

f

↦

f

⁢

(

m

)

is a left

R

-

module homomorphism

from

M

*

to

R

. In other words, the mapping is an element of

M

*

*

. We call this mapping

m

^

, since it only depends on

m

. For any

m

∈

M

, the mapping

m

↦

m

^

is a then a right

R

-module homomorphism from

M

to

M

*

*

. Let us call it

θ

.

Definition

. Let

R

,

M

, and

θ

be given as above. If

θ

is

injective

, we say that

M

is

torsionless

. If

θ

is in

addition

an

isomorphism

, we say that

M

is

reflexive

. A torsionless module is sometimes referred to as being semi-reflexive.

An obvious example of a reflexive module is any

vector space

over a field (similarly, a right vector space over a

division ring

).

Some of the properties of torsionless and reflexive modules are

•

any

free module

is torsionless.

•

any

direct sum

of torsionless modules is torsionless; any

submodule

of a torsionless module is torsionless.

•

based on the two properties above, any

projective module

is torsionless.

•

R

is reflexive.

•

any finite direct sum of reflexive modules is reflexive; any

direct summand

of a reflexive module is reflexive.

•

based on the two immediately preceding properties, any

finitely generated projective module

is reflexive.

Title

reflexive module

Canonical name

ReflexiveModule

Date of creation

2013-03-22 19:22:38

Last modified on

2013-03-22 19:22:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D90

Classification

msc 16D80

Defines

torsionless

Defines

reflexive

Generated on Fri Feb 9 18:40:02 2018 by

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Hopfian module

Hopfian module

A left (right) module

M

over a ring

R

is

Hopfian

if every

surjective

R

-

endomorphism

of

M

is an automorphism. Dually, a left (right)

R

-module

M

is

cohopfian

if every

injective

R

-endomorphism of

M

is an automorphism.

References

1

T. Y. Lam,

Lectures on Modules and Rings

, Springer-Verlag, New York (1999).

Title

Hopfian module

Canonical

name

HopfianModule

Date of creation

2013-03-22 14:20:21

Last modified on

2013-03-22 14:20:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D99

Related topic

HopfianGroup

Defines

cohopfian module

Generated on Fri Feb 9 18:40:04 2018 by

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ideal included in union of prime ideals

ideal included in union of prime ideals

In the following

R

is a

commutative ring

with unity.

Proposition 1

.

Let

I

be an ideal of the ring

R

and

P

1

,

P

2

,

…

,

P

n

be

prime ideals

of

R

. If

I

⊈

P

i

, for all

i

, then

I

⊈

∪

P

i

.

Proof.

We will prove by

induction

on

n

. For

n

=

1

the proof is

trivial

. Assume now that the result is true for

n

-

1

. That implies the

existence

, for each

i

, of an

element

s

i

such that

s

i

∈

I

and

s

i

∉

⋃

j

≠

i

P

j

. If for some

i

,

s

i

∉

P

i

then we are done. Thus, we may consider only the case

s

i

∈

P

i

, for all

i

.

Let

a

i

=

r

1

⁢

…

⁢

r

i

-

1

⁢

r

i

+

1

⁢

…

⁢

r

n

. Since

P

i

is prime then

a

i

∉

P

i

, for all

i

. Moreover, for

j

≠

i

, the element

a

i

∈

P

j

. Consider the element

a

=

∑

a

j

∈

I

. Since

a

i

=

a

-

∑

i

≠

j

a

j

and

∑

i

≠

j

a

j

∈

P

i

, it follows that

a

∉

P

i

, otherwise

a

i

∈

P

i

,

contradiction

. The existence of the element

a

proves the

proposition

.∎

Corollary 1

.

Let

I

be an ideal of the ring

R

and

P

1

,

P

2

,

…

,

P

n

be prime ideals of

R

. If

I

⊆

∪

P

i

, then

I

⊆

P

i

, for some

i

.

Title

ideal included in union of prime ideals

Canonical name

IdealIncludedInUnionOfPrimeIdeals

Date of creation

2013-03-22 16:53:14

Last modified on

2013-03-22 16:53:14

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

10

Author

polarbear (3475)

Entry type

Result

Classification

msc 16D99

Classification

msc 13C99

Synonym

prime avoidance lemma

Related topic

IdealsContainedInAUnionOfIdeals

Generated on Fri Feb 9 18:40:06 2018 by

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Lying-Over Theorem

Lying-Over Theorem

Let

𝔬

be a

subring

of a

commutative ring

𝔒

with

nonzero unity

and integral over

𝔬

.  If

𝔞

is an ideal of

𝔬

and

𝔄

an ideal of

𝔒

such that

𝔄

∩

𝔬

=

𝔞

,

then

𝔄

is said to

lie over

𝔞

.

Theorem.

If

𝔭

is a

prime ideal

of a ring

𝔬

which is a subring of a commutative ring

𝔒

with nonzero unity and integral over

𝔬

, then there exists a prime ideal

𝔓

of

𝔒

lying over

𝔭

.  If the prime ideals

𝔓

and

𝔔

both lie over

𝔭

and

𝔓

⊆

𝔔

,  then

𝔓

=

𝔔

.

References

1

M. Larsen & P. McCarthy

:

Multiplicative theory of ideals

.  Academic Press, New York (1971).

2

P.

Jaffard

:

Les systèmes d’idéaux

.  Dunod, Paris (1960).

Title

Lying-Over Theorem

Canonical name

LyingOverTheorem

Date of creation

2013-03-22 19:15:42

Last modified on

2013-03-22 19:15:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16D99

Classification

msc 13C99

Defines

lie over

Generated on Fri Feb 9 18:40:07 2018 by

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