aarjaneiro/planetmath huggingface dataset:
fundamental theorem of symmetric polynomials

fundamental theorem of symmetric polynomials

Every

symmetric polynomial

P

⁢

(

x

1

,

x

2

,

…

,

x

n

)

in the

indeterminates

x

1

,

x

2

,

…

,

x

n

can be expressed as a

polynomial

Q

⁢

(

p

1

,

p

2

,

…

,

p

n

)

in the

elementary symmetric polynomials

p

1

,

p

2

,

…

,

p

n

of

x

1

,

x

2

,

…

,

x

n

.  The polynomial

Q

is unique, its

coefficients

are

elements of

the ring determined by the coefficients of

P

and its degree with respect to

p

1

,

p

2

,

…

,

p

n

is same as the degree of

P

with respect to

x

1

.

Title

fundamental theorem of symmetric polynomials

Canonical name

FundamentalTheoremOfSymmetricPolynomials

Date of creation

2013-03-22 19:07:40

Last modified on

2013-03-22 19:07:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13B25

Classification

msc 12F10

Synonym

fundamental theorem of symmetric functions

Generated on Fri Feb 9 17:26:31 2018 by

LaTeXML
ring adjunction

ring adjunction

Let

R

be a

commutative ring

and

E

an

extension ring

of it.  If

α

∈

E

and commutes with all elements of

R

, then the smallest

subring

of

E

containing

R

and

α

is denoted by

R

⁢

[

α

]

.  We say that

R

⁢

[

α

]

is obtained from

R

by adjoining

α

to

R

via

ring adjunction

.

By the about “

evaluation
homomorphism

”,

R

⁢

[

α

]

=

{

f

⁢

(

α

)

∣

f

⁢

(

X

)

∈

R

⁢

[

X

]

}

,

where

R

⁢

[

X

]

is the

polynomial ring

in one

indeterminate

over

R

.  Therefore,

R

⁢

[

α

]

consists of all

expressions

which
can be formed of

α

and elements of the ring

R

by using

additions

,

subtractions

and

multiplications

.

Examples:

The polynomial rings

R

⁢

[

X

]

, the ring

ℤ

⁢

[

i

]

of the

Gaussian integers

, the ring

ℤ

⁢

[

-

1

+

i

⁢

3

2

]

of

Eisenstein integers

.

Title

ring adjunction

Canonical

name

RingAdjunction

Date of creation

2014-02-18 14:13:46

Last modified on

2014-02-18 14:13:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B25

Classification

msc 13B02

Related topic

GeneratedSubring

Related topic

FiniteRingHasNoProperOverrings

Related topic

GroundFieldsAndRings

Related topic

PolynomialRingOverIntegralDomain

Related topic

AConditionOfAlgebraicExtension

Related topic

IntegralClosureIsRing

Generated on Fri Feb 9 17:26:32 2018 by

LaTeXML
Schur polynomial

Schur polynomial

A

Schur polynomial

is a special

symmetric polynomial

associated
to a

partition

of an integer, or equivalently to a Young diagram.
Schur polynomials also have a

power series

generalization

, the

Schur functions

.

First we define some notation. Let

λ

be a partition of

n

,
and let

T

be a filling of the Young diagram for

λ

. Then by

x

T

we mean the

monomial

x

T

=

∏

i

=

1

∞

x

i

c

i

⁢

(

T

)

,

where

c

i

⁢

(

T

)

is the number of times the number

i

appears in the
filling

T

. Since

T

only has finitely many boxes, the

product

is
finite. For example, let

λ

=

(

3

,

3

,

2

,

2

)

, and let

T

be the
filling

Generated on Fri Feb 9 17:26:42 2018 by

LaTeXML
symmetric polynomial

symmetric polynomial

A

polynomial

f

∈

R

⁢

[

x

1

,

…

,

x

n

]

in

n

variables

with

coefficients

in a ring

R

is

symmetric

if

σ

⁢

(

f

)

=

f

for every

permutation

σ

of the set

{

x

1

,

…

,

x

n

}

.

Every

symmetric polynomial

can be written as a polynomial expression in the

elementary symmetric polynomials

.

Title

symmetric polynomial

Canonical

name

SymmetricPolynomial

Date of creation

2013-03-22 12:08:55

Last modified on

2013-03-22 12:08:55

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 13B25

Classification

msc 12F10

Generated on Fri Feb 9 17:26:44 2018 by

LaTeXML
cancellation ideal

cancellation ideal

Let

R

be a

commutative ring

containing

regular elements

and

𝔖

be the

multiplicative

semigroup

of the non-zero

fractional ideals

of

R

.  A fractional ideal

𝔞

of

R

is called a

cancellation ideal

or simply

cancellative

, if it is a cancellative element of

𝔖

, i.e. if

𝔞

⁢

𝔟

=

𝔞

⁢

𝔠

⇒

𝔟

=

𝔠

∀

𝔟

,

𝔠

∈

𝔖

.

•

Each invertible ideal is cancellative.

•

A finite product

𝔞

1

⁢

𝔞

2

⁢

…

⁢

𝔞

m

of fractional ideals is cancellative iff every

𝔞

i

is such.

•

The fractional ideal

𝔞

/

r

:=

{

a

⁢

r

-

1

:

a

∈

𝔞

}

,  where

𝔞

is an integral ideal of

R

and

r

a regular element of

R

, is cancellative if and only if

𝔞

is cancellative in the multiplicative semigroup of the non-zero integral ideals of

R

.

•

If

r

∈

R

,  then the

principal ideal

(

r

)

of

R

is cancellative if and only if

r

is a regular element of the

total ring of fractions

of

R

.

•

If

𝔞

1

+

𝔞

2

+

…

+

𝔞

m

is a cancellation ideal and

n

a

positive integer

, then

(

𝔞

1

+

𝔞

2

+

…

+

𝔞

m

)

n

=

𝔞

1

n

+

𝔞

2

n

+

…

+

𝔞

m

n

.

In particular, if the ideal

(

a

1

,

a

2

,

…

,

a

m

)

of

R

is cancellative, then

(

a

1

,

a

2

,

…

,

a

m

)

n

=

(

a

1

n

,

a

2

n

,

…

,

a

m

n

)

.

References

1

R. Gilmer:

Multiplicative ideal theory

.  Queens University Press. Kingston, Ontario (1968).

2

M. Larsen & P. McCarthy:

Multiplicative theory of ideals

.  Academic Press. New York (1971).

Title

cancellation ideal

Canonical name

CancellationIdeal

Date of creation

2015-05-06 14:49:08

Last modified on

2015-05-06 14:49:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Synonym

cancellative ideal

Related topic

CancellativeSemigroup

Related topic

IdealDecompositionInDedekindDomain

Defines

cancellative

Generated on Fri Feb 9 17:27:03 2018 by

LaTeXML
complete ring of quotients

complete ring of quotients

Consider a

commutative

unitary ring

R

and set

𝒮

:=

{

Hom

R

⁢

(

I

,

R

)

:

I

⁢

is dense in

⁢

R

}

(here

Hom

R

⁢

(

I

,

R

)

is the set of

R

-module

morphisms

from

I

to

R

) and define

A

:=

⋃

B

∈

𝒮

B

.

Now we shall assign a ring

structure

to

A

by defining its

addition

and multiplication. Given two

dense ideals

I

1

,

I

2

⊂

R

and two

elements

f

i

∈

Hom

R

⁢

(

I

i

,

R

)

for

i

∈

{

1

,

2

}

, one can easily check that

I

1

∩

I

2

and

f

2

-

1

⁢

(

I

1

)

are nontrivial (i.e. they aren’t

{

0

}

) and in fact also dense ideals so we define

f

1

+

f

2

∈

Hom

R

⁢

(

I

1

∩

I

2

,

R

)

by

(

f

1

+

f

2

)

⁢

(

x

)

=

f

1

⁢

(

x

)

+

f

2

⁢

(

x

)

f

1

*

f

2

∈

Hom

R

⁢

(

f

2

-

1

⁢

(

I

1

)

,

R

)

by

(

f

1

*

f

2

)

⁢

(

x

)

=

f

1

⁢

(

f

2

⁢

(

x

)

)

It is easy to check that

A

is in fact a

commutative ring

with unity.
The elements of

A

are called .

There is also an

equivalence relation

that one can define on

A

.
Given

f

i

∈

Hom

R

⁢

(

I

i

,

R

)

for

i

∈

{

1

,

2

}

, we write

f

1

∼

f

2

⇔

f

1

|

I

1

∩

I

2

=

f

2

|

I

1

∩

I

2

(i.e.

f

1

and

f

2

belong to the same

equivalence class

iff they agree on the

intersection

of the dense ideal where they are defined).

The

factor ring

Q

(

R

)

:=

A

/

∼

is then called the

complete ring of quotients

.

Remark

.

R

⊂

T

⁢

(

R

)

⊂

Q

⁢

(

R

)

, where

T

⁢

(

R

)

is the total quotient ring. One can also in general define complete ring of quotients on noncommutative rings.

References

Huckaba

J.A. Huckaba

,
”Commutative rings with

zero divisors

”,
Marcel Dekker 1988

Title

complete ring of quotients

Canonical name

CompleteRingOfQuotients

Date of creation

2013-03-22 16:20:29

Last modified on

2013-03-22 16:20:29

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

17

Author

jocaps (12118)

Entry type

Definition

Classification

msc 13B30

Related topic

CompleteRingOfQuotientsOfReducedCommutativeRings

Related topic

EpimorphicHull

Defines

fraction of rings

Defines

complete ring of quotients

Generated on Fri Feb 9 17:27:05 2018 by

LaTeXML
complete ring of quotients of reduced commutative rings

complete ring of quotients of reduced commutative rings

There is a

characterization

of

complete ring of quotients of reduced commutative rings

. Let

A

be a

reduced

(

http://planetmath.org/ReducedRing

)

commutative ring

, then if

B

is an

overring

of

A

and if for any

element

b

∈

B

\

{

0

}

there is an

a

∈

A

such that

a

⁢

b

∈

A

\

{

0

}

, then

B

is said to be a

rational extension

of

A

. See how

similar

this is with the definition of

essential extension

in the category of rings, obviously all rational extensions of reduced commutative rings are also essential extensions. Furthermore there is a maximum (upto

A

-

isomorphism

) rational extension of

A

and this is in fact the

complete ring of quotients

of

A

.

Title

complete ring of quotients of reduced commutative rings

Canonical name

CompleteRingOfQuotientsOfReducedCommutativeRings

Date of creation

2013-03-22 18:27:33

Last modified on

2013-03-22 18:27:33

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

6

Author

jocaps (12118)

Entry type

Theorem

Classification

msc 13B30

Related topic

CompleteRingOfQuotients

Related topic

essentialmonomorphism

Defines

rational extension

Generated on Fri Feb 9 17:27:07 2018 by

LaTeXML
extension by localization

extension by localization

Let

R

be a

commutative ring

and let

S

be a non-empty

multiplicative subset

of

R

.  Then the localisation (

http://planetmath.org/

Localization

) of

R

at

S

gives the commutative ring

S

-

1

⁢

R

but, generally, it has no

subring

isomorphic

to

R

.  Formally,

S

-

1

⁢

R

consists of all

elements

a

s

(

a

∈

R

,

s

∈

S

).  Therefore,

S

-

1

⁢

R

is called also a

ring of quotients

of

R

.  If

0

∈

S

, then

S

-

1

⁢

R

=

{

0

}

;  we assume now that

0

∉

S

.

•

The

mapping

a

↦

a

⁢

s

s

, where

s

is any element of

S

, is well-defined and a

homomorphism

from

R

to

S

-

1

⁢

R

.  All elements of

S

are mapped to units of

S

-

1

⁢

R

.

•

If, especially,

S

contains no

zero divisors

of the ring

R

, then the above mapping is an

isomorphism

from

R

to a certain subring of

S

-

1

⁢

R

, and we may think that

S

-

1

⁢

R

⊇

R

.  In this case, the

ring of fractions

of

R

is an

extension ring

of

R

; this concerns of course the case that

R

is an

integral domain

.  But if

R

is a

finite ring

, then

S

-

1

⁢

R

=

R

,  and no proper extension is obtained.

Title

extension by localization

Canonical name

ExtensionByLocalization

Date of creation

2013-03-22 14:24:42

Last modified on

2013-03-22 14:24:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Synonym

ring extension by localization

Related topic

TotalRingOfFractions

Related topic

ClassicalRingOfQuotients

Related topic

FiniteRingHasNoProperOverrings

Defines

ring of fractions

Defines

ring of quotients

Generated on Fri Feb 9 17:27:09 2018 by

LaTeXML
fraction field

fraction field

Given an

integral domain

R

, the

fraction field

of

R

is the

localization

S

-

1

⁢

R

of

R

with respect to the

multiplicative set

S

=

R

∖

{

0

}

. It is always a field.

Title

fraction field

Canonical name

FractionField

Date of creation

2013-03-22 11:50:27

Last modified on

2013-03-22 11:50:27

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 13B30

Synonym

field of fractions

Synonym

quotient field

Related topic

Localization

Related topic

RationalFunction

Generated on Fri Feb 9 17:27:11 2018 by

LaTeXML
fractional ideal of commutative ring

fractional ideal of commutative ring

Definition.

Let

R

be a

commutative ring

having a

regular element

and let

T

be the

total ring of fractions

of

R

.  An

R

-

submodule

(

http://planetmath.org/Submodule

)

𝔞

of

T

is called

fractional ideal

of

R

, provided that there exists a regular element

d

of

R

such that

𝔞

⁢

d

⊆

R

.  If a fractional ideal is contained in

R

, it is a usual ideal of

R

, and we can call it an

integral ideal

of

R

.

Note that a fractional ideal of

R

is not necessarily a

subring

of

T

.  The set of all fractional ideals of

R

form under the

multiplication

an

commutative semigroup

with

identity element

R

′

=

R

+

ℤ

⁢

e

,  where

e

is the unity of

T

.

An ideal

𝔞

( or fractional) of

R

is called

invertible

, if there exists another ideal

𝔞

-

1

of

R

such that

𝔞

⁢

𝔞

-

1

=

R

′

.  It is not hard to show that any invertible ideal

𝔞

is

finitely generated

and

regular

(

http://planetmath.org/RegularIdeal

), moreover that the

inverse ideal

𝔞

-

1

is uniquely determined (see the entry “

invertible ideal is finitely generated

(

http://planetmath.org/InvertibleIdealIsFinitelyGenerated

)”) and may be generated by the same amount of

generators

(

http://planetmath.org/GeneratorsOfInverseIdeal

) as

𝔞

.

The set of all invertible fractional ideals of

R

forms an

Abelian group

under the multiplication.  This group has a

normal subgroup

consisting of all regular principal fractional ideals; the corresponding

factor group

is called the of the ring

R

.

Note.

In the special case that the ring

R

has a unity 1,

R

itself is the

principal ideal

(1), being the identity element of the

semigroup

of fractional ideals and the group of invertible fractional ideals.  It is called the

unit ideal

.  The unit ideal is the only integral ideal containing units of the ring.

Title

fractional ideal of commutative ring

Canonical name

FractionalIdealOfCommutativeRing

Date of creation

2015-05-06 14:40:32

Last modified on

2015-05-06 14:40:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Related topic

FractionalIdeal

Related topic

GeneratorsOfInverseIdeal

Related topic

IdealClassesFormAnAbelianGroup

Defines

fractional ideal

Defines

integral ideal

Defines

invertible ideal

Defines

invertible

Defines

inverse ideal

Defines

class group

of a ring

Defines

unit ideal

Generated on Fri Feb 9 17:27:13 2018 by

LaTeXML
invertible ideal is finitely generated

invertible ideal is finitely generated

Theorem.

Let

R

be a

commutative ring

containing

regular elements

.  Every

invertible

(

http://planetmath.org/FractionalIdealOfCommutativeRing

)

fractional ideal

𝔞

of

R

is

finitely generated

and

regular

(

http://planetmath.org/RegularIdeal

), i.e. regular elements.

Proof.

Let

T

be the

total ring of fractions

of

R

and

e

the unity of

T

. We first show that the inverse ideal of

𝔞

has the unique

quotient

presentation

(

http://planetmath.org/QuotientOfIdeals

)

[

R

′

:

𝔞

]

where

R

′

:=

R

+

ℤ

⁢

e

.  If

𝔞

-

1

is an inverse ideal of

𝔞

, it means that

𝔞

⁢

𝔞

-

1

=

R

′

.  Therefore we have

𝔞

-

1

⊆

{

t

∈

T

⋮

t

𝔞

⊆

R

′

}

=

[

R

′

:

𝔞

]

,

so that

R

′

=

𝔞

𝔞

-

1

⊆

𝔞

[

R

′

:

𝔞

]

⊆

R

′

.

This implies that

𝔞

𝔞

-

1

=

𝔞

[

R

′

:

𝔞

]

,  and because

𝔞

is a

cancellation ideal

, it must that

𝔞

-

1

=

[

R

′

:

𝔞

]

, i.e.

[

R

′

:

𝔞

]

is the unique

inverse

of the ideal

𝔞

.

Since

𝔞

[

R

′

:

𝔞

]

=

R

′

,  there exist some elements

a

1

,

…

,

a

n

of

𝔞

and the elements

b

1

,

…

,

b

n

of

[

R

′

:

𝔞

]

such that

a

1

⁢

b

1

+

⋯

+

a

n

⁢

b

n

=

e

.  Then an arbitrary element

a

of

𝔞

satisfies

a

=

a

1

⁢

(

b

1

⁢

a

)

+

⋯

+

a

n

⁢

(

b

n

⁢

a

)

∈

(

a

1

,

…

,

a

n

)

because every

b

i

⁢

a

belongs to the ring

R

′

.  Accordingly,

𝔞

⊆

(

a

1

,

…

,

a

n

)

.  Since the converse inclusion is apparent, we have seen that

{

a

1

,

…

,

a

n

}

is a finite of the invertible ideal

𝔞

.

Since the elements

b

i

belong to the total ring of fractions of

R

, we can choose such a regular element

d

of

R

that each of the products

b

i

⁢

d

belongs to

R

.  Then

d

=

a

1

⁢

(

b

1

⁢

d

)

+

⋯

+

a

n

⁢

(

b

n

⁢

d

)

∈

(

a

1

,

…

,

a

n

)

=

𝔞

,

and thus the fractional ideal

𝔞

contains a regular element of

R

, which obviously is regular in

T

, too.

References

1

R. Gilmer:

Multiplicative

ideal theory

.  Queens University Press. Kingston, Ontario (1968).

Title

invertible ideal is finitely generated

Canonical name

InvertibleIdealIsFinitelyGenerated

Date of creation

2015-05-06 14:44:03

Last modified on

2015-05-06 14:44:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13B30

Related topic

InvertibilityOfRegularlyGeneratedIdeal

Generated on Fri Feb 9 17:27:15 2018 by

LaTeXML
localization

localization

Let

R

be a

commutative ring

and let

S

be a nonempty

multiplicative subset

of

R

. The

localization

of

R

at

S

is the ring

S

-

1

⁢

R

whose

elements

are

equivalence classes

of

R

×

S

under the

equivalence relation

(

a

,

s

)

∼

(

b

,

t

)

if

r

⁢

(

a

⁢

t

-

b

⁢

s

)

=

0

for some

r

∈

S

.

Addition

and

multiplication

in

S

-

1

⁢

R

are defined by:

•

(

a

,

s

)

+

(

b

,

t

)

=

(

a

⁢

t

+

b

⁢

s

,

s

⁢

t

)

•

(

a

,

s

)

⋅

(

b

,

t

)

=

(

a

⋅

b

,

s

⋅

t

)

The equivalence class of

(

a

,

s

)

in

S

-

1

⁢

R

is usually denoted

a

/

s

. For

a

∈

R

, the localization of

R

at the

minimal

multiplicative set containing

a

is written as

R

a

. When

S

is the

complement

of a

prime ideal

𝔭

in

R

, the localization of

R

at

S

is written

R

𝔭

.

Title

localization

Canonical name

Localization

Date of creation

2013-03-22 11:50:21

Last modified on

2013-03-22 11:50:21

Owner

djao (24)

Last modified by

djao (24)

Numerical id

11

Author

djao (24)

Entry type

Definition

Classification

msc 13B30

Synonym

ring of fractions

Related topic

FractionField

Generated on Fri Feb 9 17:27:18 2018 by

LaTeXML
multiplicative set

multiplicative set

Let

A

be a set on which a

multiplication

operation

⋅

:

A

×

A

⟶

A

has been defined. A

multiplicative subset

of

A

is a subset

S

⊂

A

with the

property

that

s

⋅

t

∈

S

for every

s

,

t

∈

S

.

Title

multiplicative set

Canonical

name

MultiplicativeSet

Date of creation

2013-03-22 11:50:19

Last modified on

2013-03-22 11:50:19

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 13B30

Classification

msc 16U20

Classification

msc 47H30

Classification

msc 45G05

Related topic

localization

Defines

multiplicative subset

Generated on Fri Feb 9 17:27:30 2018 by

LaTeXML
n-system

n

-system

Let

R

be a ring. A subset

S

of

R

is said to be an

n

-system if

•

S

≠

∅

, and

•

for every

x

∈

S

, there is an

r

∈

R

, such that

x

⁢

r

⁢

x

∈

S

.

n

-systems are a

generalization

of

m

-systems (

http://planetmath.org/MSystem

) in a ring. Every

m

-system is an

n

-system, but not

conversely

. For example, for any distinct

x

,

y

∈

R

, inductively define the

elements

a

0

=

x

,

and

⁢

a

i

+

1

=

a

i

⁢

y

i

⁢

a

i

for

⁢

i

=

0

,

1

,

2

,

…

.

Form the set

A

=

{

a

n

∣

n

⁢

is a non-negative integer

}

. In

addition

, inductively define

b

0

=

y

,

and

⁢

b

j

+

1

=

b

j

⁢

x

j

⁢

b

j

for

⁢

j

=

0

,

1

,

2

⁢

…

,

and form

B

=

{

b

m

∣

m

⁢

is a non-negative integer

}

. Then both

A

and

B

are

m

-systems (as well as

n

-systems). Furthermore,

S

=

A

∪

B

is an

n

-system which is not an

m

-system.

The example above suggests that, given an

n

-system

S

and any

x

∈

S

, we can “construct” an

m

-system

T

⊆

S

such that

x

∈

T

. Start with

a

0

=

x

, inductively define

a

i

+

1

=

a

i

⁢

y

i

⁢

a

i

, where the

existence

of

y

i

∈

R

such that

a

i

+

1

∈

S

is guaranteed by the fact that

S

is an

n

-system. Then the

collection

T

:=

{

a

i

∣

i

⁢

is a non-negative integer

}

is a subset of

S

that is an

m

-system. For if we pick any

a

i

and

a

j

, if

i

≤

j

, then

a

i

is both the left and right

sections

of

a

j

, meaning that there are

r

,

s

∈

R

such that

a

j

=

r

⁢

a

i

=

a

i

⁢

s

(this can be easily proved inductively). As a result,

a

i

⁢

(

s

⁢

y

j

)

⁢

a

j

=

a

j

⁢

y

j

⁢

a

j

∈

S

, and

a

j

⁢

(

y

j

⁢

r

)

⁢

a

i

=

a

j

⁢

y

j

⁢

a

j

∈

S

.

Remark

n

-systems provide another

characterization

of a

semiprime ideal

: an ideal

I

⊆

R

is semiprime iff

R

-

I

is an

n

-system.

Proof.

Suppose

I

is semiprime. Let

x

∈

R

-

I

. Then

x

⁢

R

⁢

x

⊈

I

, which means there is an element

y

∈

R

such that

x

⁢

y

⁢

x

∉

I

. So

R

-

I

is an

n

-system. Now suppose that

R

-

I

is an

n

-system. Let

x

∈

R

with the condition that

x

⁢

R

⁢

x

⊆

I

. This means

x

⁢

y

⁢

x

∈

I

for all

y

∈

R

. If

x

∈

R

-

I

, then there is some

y

∈

R

with

x

⁢

y

⁢

x

∈

R

-

I

, contradicting condition on

x

. Therefore,

x

∈

I

, and

I

is semiprime.
∎

Title

n

-system

Canonical

name

Nsystem

Date of creation

2013-03-22 17:29:29

Last modified on

2013-03-22 17:29:29

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 13B30

Classification

msc 16U20

Synonym

n-system

Related topic

MSystem

Related topic

SemiprimeIdeal

Generated on Fri Feb 9 17:27:50 2018 by

LaTeXML
overring

overring

Let

R

be a

commutative ring

having

regular elements

and let

T

be the

total ring of fractions

of

R

.  Then

R

⊆

T

.  Every

subring

of

T

containing

R

is an

overring

of

R

.

Example.

Let

p

be a

rational prime

number.  The

p

-

integral

rational numbers

(

http://planetmath.org/PAdicValuation

) are the

quotients

of two

integers

such that the

divisor

(

http://planetmath.org/Division

) is not

divisible

by

p

.  The set of all

p

-integral rationals is an overring of

ℤ

.

Title

overring

Canonical name

Overring

Date of creation

2013-03-22 14:22:33

Last modified on

2013-03-22 14:22:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 13B30

Related topic

AConditionOfAlgebraicExtension

Generated on Fri Feb 9 17:27:52 2018 by

LaTeXML