characterization of prime ideals

characterization of prime ideals

This entry gives a number of

equivalent

http://planetmath.org/node/5865

characterizations
of prime ideals

in rings of different generality.

We start with a general ring

R

.

Theorem 1

.

Let

R

be a ring and

P

⊊

R

a

two-sided ideal

. Then the
following statements are equivalent:

1.

Given (left, right or two-sided) ideals

I

,

J

of

P

such that the

product of ideals

I

⁢

J

⊆

P

, then

I

⊆

P

or

J

⊆

P

.

2.

If

x

,

y

∈

R

such that

x

⁢

R

⁢

y

⊆

P

, then

x

∈

P

or

y

∈

P

.

Proof.

•

“

1

⇒

2

”:

Let

x

,

y

∈

R

such that

x

⁢

R

⁢

y

⊆

P

. Let

(

x

)

and

(

y

)

be the
(left, right or two-sided)

ideals generated by

x

and

y

,
respectively. Then each element of the product of ideals

(

x

)

⁢

R

⁢

(

y

)

can
be expanded to a finite sum of

products

each of which contains or is a
factor of the form

±

x

⁢

r

⁢

y

for a suitable

r

∈

R

. Since

P

is an
ideal and

x

⁢

R

⁢

y

⊆

P

, it follows that

(

x

)

⁢

R

⁢

(

y

)

⊆

P

. Assuming statement

1

, we have

(

x

)

⊆

P

,

R

⊆

P

or

(

y

)

⊆

P

. But

P

⊊

R

, so we have

(

x

)

⊆

P

or

(

y

)

⊆

P

and hence

x

∈

P

or

y

∈

P

.

•

“

2

⇒

1

”:

Let

I

,

J

be (left, right or two-sided) ideals, such that the product
of ideals

I

⁢

J

⊆

P

. Now

R

⁢

J

⊆

J

or

I

⁢

R

⊆

I

(depending on what type of ideal we consider), so

I

⁢

R

⁢

J

⊆

I

⁢

J

⊆

P

. If

I

⊆

P

, nothing remains to be shown. Otherwise,
let

i

∈

I

∖

P

, then

i

⁢

R

⁢

j

⊆

P

for all

j

∈

J

. Since

i

∉

P

we have by statement

2

that

j

∈

P

for all

j

∈

J

, hence

J

⊆

P

.

∎

There are some additional properties if our ring is

commutative

.

Theorem 2

.

Let

R

a

commutative ring

and

P

⊊

R

an ideal. Then the
following statements are equivalent:

1.

Given ideals

I

,

J

of

P

such that the product of

ideals

I

⁢

J

⊆

P

, then

I

⊆

P

or

J

⊆

P

.

2.

The

quotient ring

R

/

P

is a

cancellation ring

.

3.

The set

R

∖

P

is a

subsemigroup

of the
multiplicative

semigroup

of

R

.

4.

Given

x

,

y

∈

R

such that

x

⁢

y

∈

P

, then

x

∈

P

or

y

∈

P

.

5.

The ideal

P

is

maximal

in the set of such ideals of

R

which do not

intersect

a subsemigroup

S

of the multiplicative semigroup of

R

.

Proof.

•

“

1

⇒

2

”:

Let

x

¯

,

y

¯

∈

R

/

P

be arbitrary nonzero

elements

. Let

x

and

y

be representatives of

x

¯

and

y

¯

, respectively,
then

x

∉

P

and

y

∉

P

. Since

R

is commutative, each
element of the product of ideals

(

x

)

⁢

(

y

)

can be written as a product
involving the factor

x

⁢

y

. Since

P

is an ideal, we would have

(

x

)

⁢

(

y

)

⊆

P

if

x

⁢

y

∈

P

which by statement

1

would
imply

(

x

)

⊆

P

or

(

y

)

⊆

P

in

contradiction

with

x

∉

P

and

y

∉

P

. Hence,

x

⁢

y

∉

P

and thus

x

¯

⁢

y

¯

≠

0

.

•

“

2

⇒

3

”:

Let

x

,

y

∈

R

∖

P

. Let

π

:

R

→

R

/

P

be the

canonical
projection

. Then

π

⁢

(

x

)

and

π

⁢

(

y

)

are nonzero elements of

R

/

P

. Since

π

is a

homomorphism

and due to statement

2

,

π

⁢

(

x

)

⁢

π

⁢

(

y

)

=

π

⁢

(

x

⁢

y

)

≠

0

. Therefore

x

⁢

y

∉

P

, that is

R

∖

P

is

closed under

multiplication. The associative
property is inherited from

R

.

•

“

3

⇒

4

”:

Let

x

,

y

∈

R

such that

x

⁢

y

∈

P

. If both

x

,

y

were not elements of

P

, then by statement

3

x

⁢

y

would not be an element of

P

. Therefore at least one of

x

,

y

is an element of

P

.

•

“

4

⇒

1

”:

Let

I

,

J

be ideals of

R

such that

I

⁢

J

⊆

P

. If

I

⊆

P

, nothing remains to be shown. Otherwise, let

i

∈

I

∖

P

. Then for all

j

∈

J

the product

i

⁢

j

∈

I

⁢

J

, hence

i

⁢

j

∈

P

. It
follows by statement

4

that

j

∈

P

, and therefore

J

⊆

P

.

•

“

4

⇒

5

”:

The condition

4

that the set

S

=

R

∖

P

is a multiplicative semigroup.  Now

P

is trivially the greatest ideal which does not intersect

S

.

•

“

5

⇒

4

”:

We presume that

P

is maximal of the ideals of

R

which do not intersect a semigroup

S

and that

x

⁢

y

∈

P

.  Assume the contrary of the assertion, i.e. that

x

∉

P

and

y

∉

P

.  Therefore,

P

is a

proper subset

of both

(

P

,

x

)

and

(

P

,

y

)

.  Thus the maximality of

P

implies that

(

P

,

x

)

∩

S

≠

{

}

,

(

P

,

y

)

∩

S

≠

{

}

.

So we can choose the elements

s

1

and

s

2

of

S

such that

s

1

=

p

1

+

r

1

⁢

x

+

n

1

⁢

x

,

s

2

=

p

2

+

r

2

⁢

y

+

n

2

⁢

y

,

where

p

1

,

p

2

∈

P

,

r

1

,

r

2

∈

R

and

n

1

,

n

2

∈

ℤ

.  Then we see that the product

s

1

⁢

s

2

=

(

p

1

+

r

2

⁢

y

+

n

2

⁢

y

)

⁢

p

1

+

(

r

1

⁢

x

+

n

1

⁢

x

)

⁢

p

2

+

(

r

1

⁢

r

2

+

n

2

⁢

r

1

+

n

1

⁢

r

2

)

⁢

x

⁢

y

+

(

n

1

⁢

n

2

)

⁢

x

⁢

y

would belong to the ideal

P

.  But this is impossible because

s

1

⁢

s

2

is an element of the multiplicative semigroup

S

and

P

does not intersect

S

.  Thus we can conclude that either

x

or

y

belongs to the ideal

P

.

∎

If

R

has an

identity element

1

, statements

2

and

3

of
the preceding

theorem

become

stronger

:

Theorem 3

.

Let

R

be a commutative ring with identity element

1

. Then an ideal

P

of

R

is a

prime ideal

if and only if

R

/

P

is an

integral
domain

. Furthermore,

P

is prime if and only if

R

∖

P

is a
monoid with identity element

1

with respect to the multiplication in

R

.

Proof.

Let

P

be prime, then

1

∉

P

since otherwise

P

would be equal
to

R

. Now by theorem

2

R

/

P

is a cancellation
ring. The canonical projection

π

:

R

→

R

/

P

is a homomorphism,
so

π

⁢

(

1

)

is the identity element of

R

/

P

. This in turn implies that
the semigroup

R

∖

P

is a monoid with identity element

1

.
∎

Title

characterization of prime ideals

Canonical name

CharacterizationOfPrimeIdeals

Date of creation

2013-03-22 15:22:01

Last modified on

2013-03-22 15:22:01

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

9

Author

GrafZahl (9234)

Entry type

Result

Classification

msc 13C05

Classification

msc 16D25

Synonym

characterisation of prime ideals

Related topic

Localization

Related topic

QuotientRingModuloPrimeIdeal

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