aarjaneiro/planetmath huggingface dataset:
Krull’s principal ideal theorem

Krull’s principal ideal theorem

Let

R

be a

Noetherian ring

, and

P

be a

prime ideal

minimal over a

principal ideal

(

x

)

.
Then the height (

http://planetmath.org/HeightOfAPrimeIdeal

) of

P

, that is, the

dimension

(

http://planetmath.org/KrullDimension

) of

R

P

, is less than 1.
More generally, if

P

is a minimal prime of an

ideal generated by

n

elements, the height of

P

is less than

n

.

Title

Krull’s principal ideal theorem

Canonical name

KrullsPrincipalIdealTheorem

Date of creation

2013-03-22 13:12:08

Last modified on

2013-03-22 13:12:08

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Theorem

Classification

msc 13C15

Generated on Fri Feb 9 17:28:46 2018 by

LaTeXML
Artin-Rees theorem

Artin-Rees theorem

Let

A

be a

Noetherian ring

,

𝔞

an ideal,

M

a

finitely generated module

, and

N

a

submodule

. Then there exists an integer

k

≥

1

such that for all integers

n

≥

1

we have

𝔞

n

⁢

M

∩

N

=

𝔞

n

-

k

⁢

(

𝔞

k

⁢

M

∩

N

)

.

Title

Artin-Rees theorem

Canonical name

ArtinReesTheorem

Date of creation

2013-03-22 12:41:03

Last modified on

2013-03-22 12:41:03

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

7

Author

n3o (216)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:28:48 2018 by

LaTeXML
bilinear map

bilinear map

Let

R

be a ring, and let

M

,

N

and

P

be modules over

R

.
A

function

f

:

M

×

N

→

P

is said to be a

bilinear map

if for each

b

∈

M

the function

h

:

N

→

P

defined by

h

⁢

(

y

)

=

f

⁢

(

b

,

y

)

for all

y

∈

N

is linear (

http://planetmath.org/LinearTransformation

)
(that is, an

R

-

module homomorphism

),
and for each

c

∈

N

the function

g

:

M

→

P

defined by

g

⁢

(

x

)

=

f

⁢

(

x

,

c

)

for all

x

∈

M

is linear.
Sometimes we may say that the function is

R

-

bilinear

,
.

A common case is a bilinear map

V

×

V

→

V

,
where

V

is a

vector space

over a field

K

;
the vector space with this

operation

then forms an algebra over

K

.

If

R

is a

commutative ring

, then every

R

-bilinear map

M

×

N

→

P

corresponds in a natural way to a

linear map

M

⊗

N

→

P

,
where

M

⊗

N

is the

tensor product

of

M

and

N

(over

R

).

Title

bilinear map

Canonical name

BilinearMap

Date of creation

2013-03-22 15:35:47

Last modified on

2013-03-22 15:35:47

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 13C99

Synonym

bilinear function

Synonym

bilinear operation

Synonym

bilinear mapping

Synonym

bilinear operator

Synonym

bilinear pairing

Synonym

pairing

Related topic

Multilinear

Related topic

BilinearForm

Related topic

ScalarMap

Defines

bilinear

Generated on Fri Feb 9 17:28:50 2018 by

LaTeXML
bilinearity and commutative rings

bilinearity and commutative rings

We show that a

bilinear map

b

:

U

×

V

→

W

is almost always definable only for commutative rings.
The exceptions lie only where non-trivial

commutators

act trivially on one of the
three modules.

Lemma 1

.

Let

R

be a ring and

U

,

V

and

W

be

R

-modules.
If

b

:

U

×

V

→

W

is

R

-

bilinear

then

b

is also

R

-middle linear.

Proof.

Given

r

∈

R

,

u

∈

U

and

v

∈

V

then

b

⁢

(

r

⁢

u

,

v

)

=

r

⁢

b

⁢

(

u

,

v

)

and

b

⁢

(

u

,

r

⁢

v

)

=

r

⁢

b

⁢

(

u

,

v

)

so

b

⁢

(

r

⁢

u

,

v

)

=

b

⁢

(

u

,

r

⁢

v

)

.
∎

Theorem 2

.

Let

R

be a ring and

U

,

V

and

W

be

faithful

R

-modules.
If

b

:

U

×

V

→

W

is

R

-bilinear and (left or right)

non-degenerate

,
then

R

must be

commutative

.

Proof.

We may assume that

b

is left non-degenerate.
Let

r

,

s

∈

R

. Then for all

u

∈

U

and

v

∈

V

it follows that

b

⁢

(

(

s

⁢

r

)

⁢

u

,

v

)

=

s

⁢

b

⁢

(

r

⁢

u

,

v

)

=

s

⁢

b

⁢

(

u

,

r

⁢

v

)

=

b

⁢

(

s

⁢

u

,

r

⁢

v

)

=

b

⁢

(

(

r

⁢

s

)

⁢

u

,

v

)

.

Therefore

b

⁢

(

[

s

,

r

]

⁢

u

,

v

)

=

0

, where

[

s

,

r

]

=

s

⁢

r

-

r

⁢

s

. This makes

[

s

,

r

]

⁢

u

an element of the left

radical

of

b

as it is true for all

v

∈

V

.
However

b

is non-degenerate so the radical is trivial and so

[

s

,

r

]

⁢

u

=

0

for
all

u

∈

U

. Since

U

is a faithful

R

-module this makes

[

s

,

r

]

=

0

for all

s

,

r

∈

R

. That is,

R

is commutative.
∎

Alternatively we can interpret the result in a weaker fashion as:

Corollary 3

.

Let

R

be a ring and

U

,

V

and

W

be

R

-modules.
If

b

:

U

×

V

→

W

is

R

-bilinear with

W

=

⟨

b

⁢

(

U

,

V

)

⟩

then
every element

[

R

,

R

]

acts trivially
on one of the three modules

U

,

V

or

W

.

Proof.

Suppose

[

r

,

s

]

∈

[

R

,

R

]

,

[

r

,

s

]

⁢

U

≠

0

and

[

r

,

s

]

⁢

V

≠

0

. Then we have shown

0

=

b

⁢

(

[

r

,

s

]

⁢

u

,

v

)

=

[

r

,

s

]

⁢

b

⁢

(

u

,

v

)

for all

u

∈

U

and

v

∈

V

.
As

W

=

⟨

b

⁢

(

U

,

V

)

⟩

it follows that

[

r

,

s

]

⁢

W

=

0

.
∎

Whenever a non-commutative ring is required for a biadditive map

U

×

V

→

W

it is therefore often preferable to use a

scalar map

instead.

Title

bilinearity and commutative rings

Canonical name

BilinearityAndCommutativeRings

Date of creation

2013-03-22 17:24:19

Last modified on

2013-03-22 17:24:19

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

5

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:28:53 2018 by

LaTeXML
characterization of primary ideals

characterization of primary ideals

Proposition

.

Let

R

be a

commutative ring

and

I

⊆

R

an ideal. Then

I

is

primary

if and only if every

zero divisor

in

R

/

I

is

nilpotent

.

Proof.

,,

⇒

” Assume, that we have

x

∈

R

such that

x

+

I

is a zero divisor in

R

/

I

. In particular

x

+

I

≠

0

+

I

and there is

y

∈

R

,

y

+

I

≠

0

+

I

such that

0

+

I

=

(

x

+

I

)

⁢

(

y

+

I

)

=

x

⁢

y

+

I

.

This is if and only if

x

⁢

y

∈

I

. Thus either

y

∈

I

or

x

n

∈

I

for some

n

∈

ℕ

. Of course

y

∉

I

, because

y

+

I

≠

0

+

I

and thus

x

n

∈

I

. Therefore

x

n

+

I

=

0

+

I

, which means that

x

+

I

is nilpotent in

R

/

I

.

,,

⇐

” Assume that for some

x

,

y

∈

R

we have

x

⁢

y

∈

I

and

x

,

y

∉

I

. Then

(

x

+

I

)

⁢

(

y

+

I

)

=

x

⁢

y

+

I

=

0

+

I

,

so both

x

+

I

and

y

+

I

are zero divisors in

R

/

I

. By our

assumption

both are nilpotent, and therefore there is

n

,

m

∈

ℕ

such that

x

n

+

I

=

y

m

+

I

=

0

+

I

. This shows, that

x

n

∈

I

and

y

m

∈

I

, which

completes

the proof.

□

Title

characterization of primary ideals

Canonical name

CharacterizationOfPrimaryIdeals

Date of creation

2013-03-22 19:04:29

Last modified on

2013-03-22 19:04:29

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

5

Author

joking (16130)

Entry type

Derivation

Classification

msc 13C99

Generated on Fri Feb 9 17:28:55 2018 by

LaTeXML
counter example to Nakayama’s lemma for non-finitely generated modules

counter example to Nakayama’s lemma for non-finitely generated modules

The hypothesis that the module

M

be

finitely generated

is really
necessary. For example, the field of

p

-adic numbers

ℚ

p

is
not finitely generated over its

ring of integers

ℤ

p

and

(

p

)

⁢

ℚ

p

=

ℚ

p

.

In one sense, the reason why

ℚ

p

is “bad” is that is has no
proper sub module which is also

maximal

. Thus

ℚ

p

has no non-zero simple

quotient

. This explains why the following
Proof of Nakayama’s Lemma (

http://planetmath.org/ProofOfNakayamasLemma2

)
does not work for non-finitely generated modules.

Title

counter example to Nakayama’s lemma for non-finitely generated modules

Canonical name

CounterExampleToNakayamasLemmaForNonfinitelyGeneratedModules

Date of creation

2013-03-22 18:03:55

Last modified on

2013-03-22 18:03:55

Owner

sjm (20613)

Last modified by

sjm (20613)

Numerical id

9

Author

sjm (20613)

Entry type

Example

Classification

msc 13C99

Generated on Fri Feb 9 17:28:57 2018 by

LaTeXML
definition of prime ideal by Artin

definition of prime ideal by Artin

Lemma.

Let

R

be a

commutative ring

and

S

a

multiplicative

semigroup

consisting of a subset of

R

.  If there exist

http://planetmath.org/node/371

ideals of

R

which are disjoint with

S

, then the set

𝔖

of all such ideals has a

maximal element

with respect to the set inclusion.

Proof.

Let

C

be an arbitrary chain in

𝔖

.  Then the union

𝔟

:=

⋃

𝔞

∈

C

𝔞

,

which belongs to

𝔖

, may be taken for the

upper bound

of

C

, since it clearly is an ideal of

R

and disjoint with

S

.  Because

𝔖

thus is

inductively ordered

with respect to “

⊆

”, our assertion follows from Zorn’s lemma.

Definition.

The maximal elements in the Lemma are

prime ideals

of the commutative ring.

The ring

R

itself is always a prime ideal (

S

=

∅

).  If

R

has no

zero divisors

, the

zero ideal

(

0

)

is a prime ideal (

S

=

R

∖

{

0

}

).

If the ring

R

has a

non-zero unity

element 1, the prime ideals corresponding the semigroup

S

=

{

1

}

are the

maximal ideals

of

R

.

References

1

Emil Artin

:

Theory of

Algebraic Numbers

.  Lecture notes.  Mathematisches Institut, Göttingen (1959).

Title

definition of prime ideal by Artin

Canonical name

DefinitionOfPrimeIdealByArtin

Date of creation

2013-03-22 18:44:31

Last modified on

2013-03-22 18:44:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 13C99

Classification

msc 06A06

Related topic

EveryRingHasAMaximalIdeal

Defines

prime ideal

Generated on Fri Feb 9 17:28:59 2018 by

LaTeXML
equivalent formulation of Nakayama’s lemma

equivalent formulation of Nakayama’s lemma

The following is

equivalent

to Nakayama’s lemma.

Let

A

be a ring,

M

be a finitely-generated

A

-module,

N

a

submodule

of

M

, and

𝔞

an ideal of

A

contained

in its

Jacobson radical

. Then

M

=

𝔞

⁢

M

+

N

⇒

M

=

N

.

Clearly this statement implies Nakayama’s Lemma, by setting

N

to

0

. To see that it follows from Nakayama’s Lemma, note first that by the

second isomorphism theorem

for modules,

𝔞

⁢

M

+

N

N

=

𝔞

⁢

M

𝔞

⁢

M

∩

N

and the obvious map

𝔞

⁢

M

→

𝔞

⁢

M

N

:

a

⁢

m

↦

a

⁢

(

m

+

N

)

is

surjective

; the kernel is clearly

𝔞

⁢

M

∩

N

. Thus

𝔞

⁢

M

+

N

N

≅

𝔞

⁢

M

N

So from

M

=

𝔞

⁢

M

+

N

we get

M

/

N

=

𝔞

⁢

(

M

/

N

)

. Since

𝔞

is contained in the Jacobson radical of

M

, it is contained in the Jacobson radical of

M

/

N

, so by Nakayama,

M

/

N

=

0

, i.e.

M

=

N

.

Title

equivalent formulation of Nakayama’s lemma

Canonical name

EquivalentFormulationOfNakayamasLemma

Date of creation

2013-03-22 19:11:47

Last modified on

2013-03-22 19:11:47

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:29:00 2018 by

LaTeXML
has a rank

has a rank

Let

R

be a

Noetherian ring

with total quotient ring

Quot

⁡

(

R

)

, and

M

a

finitely generated

R

-module. We say

M

has a rank if

M

⊗

R

Quot

(

R

)

≅

Quot

(

R

)

n

for some non-negative integer

n

. And in this situation, we say

M

has rank

n

.

Title

has a rank

Canonical name

HasARank

Date of creation

2013-03-22 18:13:41

Last modified on

2013-03-22 18:13:41

Owner

yshen (21076)

Last modified by

yshen (21076)

Numerical id

6

Author

yshen (21076)

Entry type

Definition

Classification

msc 13C99

Generated on Fri Feb 9 17:29:02 2018 by

LaTeXML
ideals with maximal radicals are primary

ideals with maximal radicals are primary

Proposition

.

Assume that

R

is a

commutative ring

and

I

⊆

R

is an ideal, such that the

radical

r

⁢

(

I

)

of

I

is a

maximal ideal

. Then

I

is a

primary ideal

.

Proof.

We will show, that every

zero divisor

in

R

/

I

is

nilpotent

(please, see parent object for details).

First of all, recall that

r

⁢

(

I

)

is an

intersection

of all

prime ideals

containing

I

(please, see this entry (

http://planetmath.org/ACharacterizationOfTheRadicalOfAnIdeal

) for more details). Since

r

⁢

(

I

)

is maximal, it follows that there is exactly one prime ideal

P

=

r

⁢

(

I

)

such that

I

⊆

P

. In particular the ring

R

/

I

has only one prime ideal (because there is

one-to-one correspondence

between prime ideals in

R

/

I

and prime ideals in

R

containing

I

). Thus, in

R

/

I

an ideal

r

⁢

(

0

)

is prime.

Now assume that

α

∈

R

/

I

is a zero divisor. In particular

α

≠

0

+

I

and for some

β

≠

0

+

I

∈

R

/

I

we have

α

⁢

β

=

0

+

I

.

But

0

+

I

∈

r

⁢

(

0

)

and

r

⁢

(

0

)

is prime. This shows, that either

α

∈

r

⁢

(

0

)

or

β

∈

r

⁢

(

0

)

.

Obviously

α

∈

r

⁢

(

0

)

(and

β

∈

r

⁢

(

0

)

), because

r

⁢

(

0

)

is the only maximal ideal in

R

/

I

(the ring

R

/

I

is local). Therefore

elements

not belonging to

r

⁢

(

0

)

are

invertible

, but

α

cannot be invertible, because it is a zero divisor.

On the other hand

r

⁢

(

0

)

=

{

x

+

I

∈

R

/

I

|

(

x

+

I

)

n

=

0

⁢

for some

⁢

n

∈

ℕ

}

. Therefore

α

is nilpotent and this

completes

the proof.

□

Corollary.

Let

p

∈

ℕ

be a

prime number

and

n

∈

ℕ

. Then the ideal

(

p

n

)

⊆

ℤ

is primary.

Proof.

Of course the ideal

(

p

)

is maximal and we have

r

⁢

(

(

p

n

)

)

=

r

⁢

(

(

p

)

n

)

=

(

p

)

,

since for any prime ideal

P

(in arbitrary ring

R

) we have

r

⁢

(

P

n

)

=

P

. The result follows from the proposition.

□

Title

ideals with maximal radicals are primary

Canonical

name

IdealsWithMaximalRadicalsArePrimary

Date of creation

2013-03-22 19:04:31

Last modified on

2013-03-22 19:04:31

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:29:04 2018 by

LaTeXML
Nakayama’s lemma

Nakayama’s lemma

Let

R

be a

commutative ring

with 1. Let

M

be a

finitely generated

R

-module. If there exists an ideal

𝔞

of

R

contained in the

Jacobson radical

and such that

𝔞

⁢

M

=

M

, then

M

=

0

.

Title

Nakayama’s lemma

Canonical name

NakayamasLemma

Date of creation

2013-03-22 13:07:41

Last modified on

2013-03-22 13:07:41

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

6

Author

n3o (216)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:29:16 2018 by

LaTeXML
primary decomposition

primary decomposition

Let

R

be a

commutative ring

and

A

be an ideal in

R

. A

decomposition

of

A

is a way of writing

A

as a finite

intersection

of

primary ideals

:

A

=

⋂

i

=

1

n

Q

i

,

where the

Q

i

are primary in

R

.

Not every ideal admits a

primary decomposition

, so we define a

decomposable ideal

to be one that does.

Example

. Let

R

=

ℤ

and take

A

=

(

180

)

. Then

A

is

decomposable

, and a primary decomposition of

A

is given by

A

=

(

4

)

∩

(

9

)

∩

(

5

)

,

since

(

4

)

,

(

9

)

, and

(

5

)

are all primary ideals in

ℤ

.

Given a primary decomposition

A

=

∩

Q

i

, we say that the decomposition is a

minimal primary decomposition

if for all

i

, the

prime ideals

P

i

=

rad

⁢

(

Q

i

)

(where rad denotes the

radical

of an ideal) are distinct, and for all

1

≤

i

≤

n

, we have

Q

i

⊄

⋂

j

≠

i

Q

j

In the example above, the decomposition

(

4

)

∩

(

9

)

∩

(

5

)

of

A

is

minimal

, where as

A

=

(

2

)

∩

(

4

)

∩

(

3

)

∩

(

9

)

∩

(

5

)

is not.

Every primary decomposition can be refined to admit a minimal primary decomposition.

Title

primary decomposition

Canonical name

PrimaryDecomposition

Date of creation

2013-03-22 14:15:05

Last modified on

2013-03-22 14:15:05

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13C99

Synonym

shortest primary decomposition

Defines

decomposable ideal

Defines

minimal primary decomposition

Generated on Fri Feb 9 17:29:18 2018 by

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primary decomposition theorem

primary decomposition theorem

The

primary decomposition theorem

for ideals in a given

commutative ring

(with 1) is a

generalization

of the

fundamental theorem of arithmetic

. The full statement of the

theorem

is as follows:

Theorem 1

.

Every

decomposable ideal

in a commutative ring

R

with 1 has a unique minimal primary decomposition. In other words, if

I

is an ideal of

R

with two minimal primary decompositions

I

=

J

1

∩

J

2

∩

⋯

∩

J

m

=

K

1

∩

K

2

∩

⋯

∩

K

n

,

then

m

=

n

, and after some rearrangement,

rad

⁡

(

J

i

)

=

rad

⁡

(

K

i

)

.

The theorem says, that, the number of primary components of a minimal primary decomposition of an ideal, as well as the set of

prime radicals

associated with the primary components, are unique. This is not to say, however, that the ideal has a unique minimal primary decomposition. For example, let

k

be a field. Consider the ring

k

⁢

[

x

,

y

]

of

polynomials

over

k

in two

variables

. The ideal

(

x

2

,

x

⁢

y

)

has minimal primary decompositions

(

x

)

∩

(

x

2

,

y

+

r

⁢

x

)

for every

r

∈

k

.

Remark

. To tie the fundamental theorem of arithmetic with this theorem, we observe that every

natural number

n

greater than

1

can be uniquely expressed as a

product

of prime powers:

n

=

p

1

a

1

⁢

p

2

a

2

⁢

⋯

⁢

p

n

a

n

,

where each

p

i

is a

prime number

. This is the same as saying that

(

n

)

=

(

p

1

a

1

)

∩

(

p

2

a

2

)

∩

⋯

∩

(

p

n

a

n

)

,

as every

(

p

a

)

is a

(

p

)

-

primary ideal

of

ℤ

for every prime

p

∈

ℕ

. The

decomposition

is

minimal

, as

(

p

1

a

1

)

=

(

p

2

a

2

)

iff

p

1

=

p

2

and

a

1

=

a

2

.

References

1

D.G. Northcott,

Ideal Theory

, Cambridge University Press, 1953.

Title

primary decomposition theorem

Canonical name

PrimaryDecompositionTheorem1

Date of creation

2013-03-22 18:19:56

Last modified on

2013-03-22 18:19:56

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 13C99

Generated on Fri Feb 9 17:29:28 2018 by

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primary ideal

primary ideal

An ideal

Q

in a

commutative ring

R

is a

primary ideal

if for all

elements

x

,

y

∈

R

, we have that if

x

⁢

y

∈

Q

, then either

x

∈

Q

or

y

n

∈

Q

for some

n

∈

ℕ

.

This is clearly a

generalization

of the notion of a

prime ideal

, and (very) loosely mirrors the relationship in

ℤ

between prime numbers and prime powers.

It is clear that every prime ideal is primary.

Example.

Let

Q

=

(

25

)

in

R

=

ℤ

. Suppose that

x

⁢

y

∈

Q

but

x

∉

Q

. Then

25

|

x

y

, but 25 does not divide

x

. Thus 5 must divide

y

, and thus some power of

y

(namely,

y

2

), must be in

Q

.

The

radical

of a primary ideal is always a prime ideal. If

P

is the radical of the primary ideal

Q

, we say that

Q

is

P

-primary

.

Title

primary ideal

Canonical name

PrimaryIdeal

Date of creation

2013-03-22 14:15:01

Last modified on

2013-03-22 14:15:01

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 13C99

Defines

primary

Defines

P

-primary

Generated on Fri Feb 9 17:29:30 2018 by

LaTeXML
prime element

prime element

An

element

p

in a ring

R

is a

prime element

if it

generates

a

prime ideal

. If

R

is

commutative

, this is

equivalent

to saying that for all

a

,

b

∈

R

, if

p

divides

a

⁢

b

, then

p

divides

a

or

p

divides

b

.

When

R

=

ℤ

the prime elements as formulated above are simply prime numbers.

Title

prime element

Canonical name

PrimeElement

Date of creation

2013-03-22 12:46:52

Last modified on

2013-03-22 12:46:52

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 13C99

Classification

msc 16D99

Synonym

prime

Related topic

PrimeIdeal

Related topic

DivisibilityInRings

Related topic

DivisibilityByPrimeNumber

Generated on Fri Feb 9 17:29:31 2018 by

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prime ideals by Artin are prime ideals

prime ideals by Artin are prime ideals

Theorem

.

Due to Artin, a

prime ideal

of a

commutative ring

R

is the

maximal element

among the ideals not intersecting a

multiplicative subset

S

of

R

.  This is

equivalent

(

http://planetmath.org/Equivalent3

) to the usual criterion

a

⁢

b

∈

𝔭

⇒

a

∈

𝔭

∨

b

∈

𝔭

(1)

of prime ideal (see the entry prime ideal (

http://planetmath.org/PrimeIdeal

)).

Proof.

1

o

¯

.  Let

𝔭

be a prime ideal by Artin, corresponding the

semigroup

S

, and let the

ring product

a

⁢

b

belong to

𝔭

.  Assume, contrary to the assertion, that  neither of

a

and

b

lies in

𝔭

.  When

(

𝔭

,

x

)

generally means the least ideal containing

𝔭

and an

element

x

, the antithesis implies that

𝔭

⊂

(

𝔭

,

a

)

∧

𝔭

⊂

(

𝔭

,

a

)

,

whence by the maximality of

𝔭

we have

(

𝔭

,

a

)

∩

S

≠

∅

∧

(

𝔭

,

b

)

∩

S

≠

∅

.

Therefore we can chose such elements

s

i

=

p

i

+

r

i

⁢

a

+

n

i

⁢

a

of

S

(N.B. the

multiples

) that

p

i

∈

𝔭

,

r

i

∈

R

,

n

i

∈

ℤ

(

i

=

1

,

2

)

.

But then

s

1

⁢

s

2

=

(

p

2

+

r

2

⁢

b

+

n

2

⁢

b

)

⁢

p

1

+

(

r

1

⁢

a

+

n

1

⁢

a

)

⁢

p

2

+

(

r

1

⁢

r

2

+

n

2

⁢

r

1

+

n

1

⁢

r

2

)

⁢

a

⁢

b

+

(

n

1

⁢

n

2

)

⁢

a

⁢

b

∈

𝔭

.

This is however impossible, since the

product

s

1

⁢

s

2

belongs to the semigroup

S

and

𝔭

∩

S

=

∅

.  Because the antithesis thus is wrong, we must have

a

∈

𝔭

or

b

∈

𝔭

.

2

o

¯

.  Let us then suppose that an ideal

𝔭

satisfies

the condition (1) for all

a

,

b

∈

R

.  It means that the set

S

=

R

∖

𝔭

is a multiplicative semigroup.  Accordingly, the

𝔭

is the greatest ideal not intersecting the semigroup

S

, Q.E.D.

Remark.

It follows easily from the theorem, that if

𝔭

is a prime ideal of the commutative ring

𝔒

and

𝔬

is a

subring

of

𝔒

, then

𝔭

∩

𝔬

is a prime ideal of

𝔬

.

Title

prime ideals by Artin are prime ideals

Canonical name

PrimeIdealsByArtinArePrimeIdeals

Date of creation

2013-03-22 18:44:55

Last modified on

2013-03-22 18:44:55

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13C99

Classification

msc 06A06

Related topic

IdealGeneratedByASet

Related topic

PrimeIdeal

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