aarjaneiro/planetmath huggingface dataset:
length of a module

length of a module

Let

A

be a ring and let

M

be an

A

-module. If there is a finite sequence of

submodules

of

M

M

=

M

0

⊃

M

1

⊃

⋯

⊃

M

n

=

0

such that each

quotient module

M

i

/

M

i

+

1

is simple, then

n

is necessarily unique by the Jordan-Hölder theorem (

http://planetmath.org/JordanHolderDecomposition

) for modules. We define the above number

n

to be the

length

of

M

. If such a finite sequence does not exist, then the length of

M

is defined to be

∞

.

If

M

has

finite length

, then

M

satisfies both the ascending and

descending chain conditions

.

A ring

A

is said to have

finite length

if there is an

A

-module whose length is finite.

Title

length of a module

Canonical name

LengthOfAModule

Date of creation

2013-03-22 14:35:32

Last modified on

2013-03-22 14:35:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D10

Classification

msc 13C15

Synonym

finite-length module

Defines

finite length

Generated on Fri Feb 9 18:36:24 2018 by

LaTeXML
modular law

modular law

Let

M

R

be a left

R

-module with submodules

A

,

B

,

C

, and suppose

C

⊆

B

. Then

C

+

(

B

∩

A

)

=

B

∩

(

C

+

A

)

This result shows that the submodules of

M

R

, partially ordered by

inclusion

, form a

modular lattice

with

∩

as the meet and

+

as the join (

http://planetmath.org/Join

).

Title

modular law

Canonical name

ModularLaw

Date of creation

2013-03-22 12:50:42

Last modified on

2013-03-22 12:50:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Theorem

Classification

msc 16D10

Related topic

ModularLattice

Generated on Fri Feb 9 18:36:26 2018 by

LaTeXML
module

module

(This is a definition of modules in terms of

ring homomorphisms

. You may prefer to read the other definition (

http://planetmath.org/Module

) instead.)

Let

R

be a ring,
and let

M

be an

abelian group

.

We say that

M

is a

left

R

-module

if there exists a ring homomorphism

ϕ

:

R

→

End

ℤ

⁢

(

M

)

from

R

to the ring of abelian group

endomorphisms

on

M

(in which

multiplication

of endomorphisms is

composition

,
using

left function notation

).
We typically denote this function using a multiplication notation:

[

ϕ

⁢

(

r

)

]

⁢

(

m

)

=

r

⋅

m

=

r

⁢

m

.

This ring homomorphism defines
what is called a
of

R

upon

M

.

If

R

is a

unital ring

(i.e. a

ring with identity

),
then we typically demand
that the ring homomorphism
map the unit

1

∈

R

to the

identity

endomorphism on

M

,
so that

1

⋅

m

=

m

for all

m

∈

M

.
In this case we may say
that the module is

unital

.

Typically the abelian group

structure

on

M

is expressed in

additive

terms,
i.e. with

operator

+

,

identity element

0

M

(or just

0

),
and

inverses

written in
the form

-

m

for

m

∈

M

.

Right module actions

are defined similarly,
only with the elements of

R

being written
on the right sides of elements of

M

.
In this case we either need to use
an

anti-homomorphism

R

→

End

ℤ

⁡

(

M

)

,
or switch to

right notation

for writing functions.

Title

module

Canonical name

Module1

Date of creation

2013-03-22 12:01:51

Last modified on

2013-03-22 12:01:51

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Definition

Classification

msc 16D10

Synonym

module action

Synonym

left module action

Synonym

right module action

Synonym

unital module

Related topic

Module

Generated on Fri Feb 9 18:36:28 2018 by

LaTeXML
module-finite extensions are integral

module-finite extensions are integral

Theorem

Suppose

B

⊂

A

is

module-finite

. Then

A

is integral over

B

.

Proof.

Choose

u

∈

A

.

For clarity, assume

A

is spanned by two elements

ω

1

,

ω

2

. The proof given clearly generalizes to the case where a

spanning set

for

A

has more than two elements.

Write

u

⁢

ω

1

=

b

11

⁢

ω

1

+

b

12

⁢

ω

2

u

⁢

ω

2

=

b

21

⁢

ω

1

+

b

22

⁢

ω

2

Consider

C

=

(

u

-

b

11

-

b

12

-

b

21

u

-

b

22

)

and let

C

adj

be the

adjugate

of

C

.
Then

C

⁢

(

ω

1

ω

2

)

=

0

, so

C

adj

⁢

C

⁢

(

ω

1

ω

2

)

=

0

.

Now,

C

adj

⁢

C

is a

diagonal matrix

with

det

⁡

C

on the

diagonal

, so

(

f

⁢

(

u

)

0

0

f

⁢

(

u

)

)

⁢

(

ω

1

ω

2

)

=

(

0

0

)

where

f

∈

B

⁢

[

x

]

is monic.

But neither

ω

1

nor

ω

2

is zero, so

f

⁢

(

u

)

must be.

Note that, as with the field case, the converse is not true. For example, the

algebraic integers

are

integral

but not finite over

ℤ

.

Title

module-finite extensions are integral

Canonical name

ModulefiniteExtensionsAreIntegral

Date of creation

2013-03-22 17:01:23

Last modified on

2013-03-22 17:01:23

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

9

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 16D10

Classification

msc 13C05

Classification

msc 13B02

Related topic

RingFiniteIntegralExtensionsAreModuleFinite

Generated on Fri Feb 9 18:36:30 2018 by

LaTeXML
Noetherian and Artinian properties are inherited in short exact sequences

Noetherian and Artinian properties are inherited in short exact sequences

Theorem 1

.

Let

M

,

M

′

,

M

′′

be

A

-modules and

0

→

M

′

⁢

→

𝜄

⁢

M

⁢

→

𝜋

⁢

M

′′

→

0

a

short exact sequence

. Then

1.

M

is

Noetherian

if and only if

M

′

and

M

′′

are Noetherian;

2.

M

is

Artinian

if and only if

M

′

and

M

′′

are Artinian.

For

⇐

, we will need a lemma that essentially says that a

submodule

of

M

is uniquely determined by its image in

M

′′

and its

intersection

with

M

′

:

Lemma 1

.

In the situation of the theorem, if

N

1

,

N

2

⊂

M

are submodules with

N

1

⊂

N

2

,

π

⁢

(

N

1

)

=

π

⁢

(

N

2

)

, and

N

1

∩

ι

⁢

(

M

′

)

=

N

2

∩

ι

⁢

(

M

′

)

, then

N

1

=

N

2

.

Proof.

The proof is essentially a

diagram

chase.
Choose

x

∈

N

2

. Then

π

⁢

(

x

)

=

π

⁢

(

x

′

)

for some

x

′

∈

N

1

, and thus

π

⁢

(

x

-

x

′

)

=

0

, so that

x

-

x

′

∈

im

⁡

ι

, and

x

-

x

′

∈

N

2

since

N

1

⊂

N

2

. Hence

x

-

x

′

∈

N

2

∩

ι

⁢

(

M

′

)

=

N

1

∩

ι

⁢

(

M

′

)

⊂

N

1

. Since

x

′

∈

N

1

, it follows that

x

∈

N

1

so that

N

1

=

N

2

.
∎

Proof.

(

⇒

): If

M

is Noetherian (Artinian), then any ascending (descending) chain of submodules of

M

′

(or of

M

′′

) gives rise to a similar sequence in

M

, which must therefore terminate. So the original chain terminates as well.

(

⇐

): Assume first that

M

′

,

M

′′

are Noetherian, and choose any ascending chain

M

1

⊂

M

2

⊂

…

of submodules of

M

. Then the ascending chain

π

⁢

(

M

1

)

⊂

π

⁢

(

M

2

)

⊂

…

and the ascending chain

M

1

∩

ι

⁢

(

M

′

)

⊂

M

2

∩

ι

⁢

(

M

′

)

⊂

…

both stabilize since

M

′

and

M

′′

are Noetherian. We can choose

n

large enough so that both chains stabilize at

n

. Then for

N

≥

n

, we have (by the lemma) that

M

N

=

M

n

since

π

⁢

(

M

N

)

=

π

⁢

(

M

n

)

and

M

N

∩

ι

⁢

(

M

′

)

=

M

n

∩

ι

⁢

(

M

′

)

. Thus

M

is Noetherian. For the case where

M

is Artinian, an identical proof applies, replacing ascending chains by descending chains.
∎

References

1

M.F. Atiyah, I.G. MacDonald,

Introduction to Commutative Algebra

, Addison-Wesley 1969.

Title

Noetherian and Artinian properties are inherited in short exact sequences

Canonical name

NoetherianAndArtinianPropertiesAreInheritedInShortExactSequences

Date of creation

2013-03-22 19:11:52

Last modified on

2013-03-22 19:11:52

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 16D10

Generated on Fri Feb 9 18:36:33 2018 by

LaTeXML
proof of modular law

proof of modular law

First we show

C

+

(

B

∩

A

)

⊆

B

∩

(

C

+

A

)

:

Note that

C

⊆

B

,

B

∩

A

⊆

B

, and therefore

C

+

(

B

∩

A

)

⊆

B

.

Further,

C

⊆

C

+

A

,

B

∩

A

⊆

C

+

A

, thus

C

+

(

B

∩

A

)

⊆

C

+

A

.

Next we show

B

∩

(

C

+

A

)

⊆

C

+

(

B

∩

A

)

:

Let

b

∈

B

∩

(

C

+

A

)

. Then

b

=

c

+

a

for some

c

∈

C

and

a

∈

A

.
Hence

a

=

b

-

c

, and so

a

∈

B

since

b

∈

B

and

c

∈

C

⊆

B

.

Hence

a

∈

B

∩

A

, so

b

=

c

+

a

∈

C

+

(

B

∩

A

)

.

Title

proof of modular law

Canonical name

ProofOfModularLaw

Date of creation

2013-03-22 12:50:45

Last modified on

2013-03-22 12:50:45

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Proof

Classification

msc 16D10

Related topic

FirstIsomorphismTheorem

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

LaTeXML
quotient module

quotient module

Let

M

be a module over a ring

R

, and let

S

be a

submodule

of

M

.
The

quotient module

M

/

S

is the

quotient group

M

/

S

with

scalar multiplication

defined by

λ

⁢

(

x

+

S

)

=

λ

⁢

x

+

S

for all

λ

∈

R

and all

x

∈

M

.

This is a well defined operation. Indeed, if

x

+

S

=

x

′

+

S

then for
some

s

∈

S

we have

x

′

=

x

+

s

and therefore

λ

⁢

x

′

=

λ

⁢

(

x

+

s

)

=

λ

⁢

x

+

λ

⁢

s

so that

λ

⁢

x

′

+

S

=

λ

⁢

x

+

λ

⁢

s

+

S

=

λ

⁢

x

+

S

,
since

λ

⁢

s

∈

S

.

In the special case that

R

is a field this construction defines
the

quotient vector space

of a

vector space

by a

vector subspace

.

Title

quotient module

Canonical name

QuotientModule

Date of creation

2013-03-22 14:01:18

Last modified on

2013-03-22 14:01:18

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16D10

Defines

quotient vector space

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

LaTeXML
ring-finite integral extensions are module-finite

ring-finite integral extensions are module-finite

Theorem

If

B

is a

subring

of

A

and

u

1

,

…

,

u

s

∈

A

are integral over

B

, then

B

⁢

[

u

1

,

…

,

u

s

]

is

module-finite

over

B

.

Proof.

If

s

=

1

then

u

n

+

b

1

⁢

u

n

-

1

+

⋯

+

b

n

=

0

, so

{

1

,

u

,

…

,

u

n

-

1

}

spans

B

⁢

[

u

]

over

B

.

If

s

>

1

, use induction on

B

⊂

B

⁢

[

u

1

]

⊂

B

⁢

[

u

1

,

u

2

]

⊂

…

⊂

B

⁢

[

u

1

,

…

,

u

s

]

and multiply the

spanning sets

together.

Title

ring-finite integral extensions are module-finite

Canonical name

RingfiniteIntegralExtensionsAreModulefinite

Date of creation

2013-03-22 17:01:28

Last modified on

2013-03-22 17:01:28

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 16D10

Classification

msc 13C05

Classification

msc 13B02

Related topic

ModuleFiniteExtensionsAreIntegral

Generated on Fri Feb 9 18:36:42 2018 by

LaTeXML
span

span

The

span

of a set of vectors

𝐯

→

1

,

…

,

𝐯

𝐧

→

of a

vector space

V

over a field

K

is the set of

linear combinations

a

1

⁢

𝐯

→

1

+

…

+

a

n

⁢

𝐯

→

n

with

a

i

∈

K

. It is denoted

Sp

⁡

(

𝐯

→

1

,

…

,

𝐯

→

n

)

. More generally, the span of a set

S

(not necessarily finite) of vectors is the collection of all (finite) linear combinations of elements of

S

. The span of the empty set is defined to be the

singleton

consisting of the

zero vector

𝟎

→

.

For example, the

standard basis vectors

ı

^

and

ȷ

^

span

ℝ

2

because every vector of

ℝ

2

can be represented as a linear combination of

ı

^

and

ȷ

^

.

Sp

⁡

(

𝐯

→

1

,

…

,

𝐯

→

n

)

is a

subspace

of

V

and is the smallest subspace containing

𝐯

→

1

,

…

,

𝐯

→

n

.

Span

is both a noun and a verb; a

set of vectors

can span a vector space, and a vector can be

in the span

of a set of vectors.

Checking span:

To see whether a vector is

in the span

of other vectors, one can set up an augmented matrix, since if

𝐮

→

is in the span of

𝐯

→

1

,

𝐯

→

2

, then

𝐮

→

=

x

1

⁢

𝐯

→

1

+

x

2

⁢

𝐯

→

2

. This is a

system of linear equations

. Thus, if it has a solution,

𝐮

→

is in the span of

𝐯

→

1

,

𝐯

→

2

. Note that the solution does not have to be unique for

𝐮

→

to be in the span.

To see whether a set of vectors

spans

a vector space, you need to check that there are at least as many

linearly independent

vectors as the

dimension

of the space. For example, it can be shown that in

ℝ

n

,

n

+

1

vectors are never linearly independent, and

n

-

1

vectors never span.

Remark

. We can define the concept of span also for a module

M

over a ring

R

.
Given a subset

X

⊂

M

we define the module generated by

X

as the set of all finite linear combinations of elements of

X

.
Be aware that in general there does not exist a linearly independent subset which generates the whole module, i.e. there does not have to exist a basis.
Also, even if

M

is generated by

n

elements, it is in general not true that any other set of

n

linearly independent elements of

M

spans

M

.
For example

ℤ

is generated by

1

as a

ℤ

-module but not by

2

.

Title

span

Canonical

name

Span

Date of creation

2013-03-22 11:58:18

Last modified on

2013-03-22 11:58:18

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

22

Author

mathwizard (128)

Entry type

Definition

Classification

msc 16D10

Classification

msc 15A03

Synonym

linear span

Related topic

LinearCombination

Related topic

Basis

Related topic

ProofOfGramSchmidtOrthogonalizationProcedure

Related topic

FinitelyGeneratedRModule

Defines

spanning set

Generated on Fri Feb 9 18:36:45 2018 by

LaTeXML
support

support

The

support

Supp

⁡

(

M

)

of a module

M

over a ring

R

is the set of all

prime ideals

𝔭

⊂

R

such that the

localization

M

𝔭

is nonzero.

The

maximal support

Supp

m

⁡

(

M

)

of a module

M

over a ring

R

is the set of all

maximal ideals

𝔪

⊂

R

such that

M

𝔪

is nonzero.

Title

support

Canonical name

Support

Date of creation

2013-03-22 12:10:22

Last modified on

2013-03-22 12:10:22

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 16D10

Classification

msc 13C99

Generated on Fri Feb 9 18:36:47 2018 by

LaTeXML
zero module

zero module

Let

R

be a ring.

The

abelian group

which contains only an

identity element

(zero)
gains a trivial

R

-module structure,
which we call the .

Every

R

-module

M

has an

zero element

and thus a

submodule

consisting of that element.
This is called the

zero submodule

of

M

.

Title

zero module

Canonical name

ZeroModule

Date of creation

2013-03-22 12:01:42

Last modified on

2013-03-22 12:01:42

Owner

antizeus (11)

Last modified by

antizeus (11)

Numerical id

6

Author

antizeus (11)

Entry type

Definition

Classification

msc 16D10

Synonym

zero submodule

Related topic

ZeroIdeal

Generated on Fri Feb 9 18:36:48 2018 by

LaTeXML
bimodule

bimodule

Let

R

and

S

be rings. An

(R,S)-

bimodule

is
an

abelian group

M

which is a

left module

over

R

and a right
module over

S

such that the

r

(

ms

)=(

rm

)

s

holds
for each

r

in

R

,

m

in

M

, and

s

in

S

.
Equivalently,

M

is an (

R

,

S

)-bimodule if it is a left
module over

R

⊗

S

op

or a right module over

R

op

⊗

S

.

When

M

is an (

R

,

S

)-bimodule, we sometimes indicate this
by writing the module as

M

S

R

.

If

P

is a

subgroup

of

M

which is also an
(

R

,

S

)-bimodule, then

P

is an

(R,S)-subbimodule

of

M

.

Title

bimodule

Canonical name

Bimodule

Date of creation

2013-03-22 12:01:18

Last modified on

2013-03-22 12:01:18

Owner

mps (409)

Last modified by

mps (409)

Numerical id

9

Author

mps (409)

Entry type

Definition

Classification

msc 16D20

Synonym

sub-bimodule

Defines

subbimodule

Generated on Fri Feb 9 18:36:50 2018 by

LaTeXML
associated prime

associated prime

Let

R

be a ring,
and let

M

be an

R

-module.
A

prime ideal

P

of

R

is an for

M

if

P

=

ann

⁢

(

X

)

, the

annihilator

of some nonzero

submodule

X

of

M

.

Note that if this is the case, then the module

ann

M

⁢

(

P

)

contains

X

, has

P

as its annihilator,
and is a

faithful

(

http://planetmath.org/FaithfulModule

)

(

R

/

P

)

-module.

If, in addition,

P

is equal to the annihilator of a submodule of

M

that is a fully faithful (

http://planetmath.org/FaithfulModule

)

(

R

/

P

)

-module, then we call

P

an of

M

.

Title

associated prime

Canonical name

AssociatedPrime

Date of creation

2013-03-22 12:01:37

Last modified on

2013-03-22 12:01:37

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 16D25

Synonym

annihilator prime

Generated on Fri Feb 9 18:37:03 2018 by

LaTeXML
comaximal ideals

comaximal ideals

Let

R

be a ring.

Two ideals

I

and

J

of

R

are said to be

comaximal

if

I

+

J

=

R

.
If

R

is unital (

http://planetmath.org/Ring

), this is

equivalent

to requiring that
there be

x

∈

I

and

y

∈

J

such that

x

+

y

=

1

.

For example, any two distinct

maximal ideals

of

R

are comaximal.

A set

𝒮

of ideals of

R

is said to be

pairwise comaximal

(or just

comaximal

) if

I

+

J

=

R

for all distinct

I

,

J

∈

𝒮

.

Title

comaximal ideals

Canonical name

ComaximalIdeals

Date of creation

2013-03-22 12:35:57

Last modified on

2013-03-22 12:35:57

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 16D25

Related topic

MaximalIdeal

Defines

comaximal

Generated on Fri Feb 9 18:37:05 2018 by

LaTeXML
criterion for maximal ideal

criterion for maximal ideal

Theorem.

In a

commutative ring

R

with

non-zero unity

, an ideal

𝔪

is

maximal

if and only if

∀

a

∈

R

∖

𝔪

⁢

∃

r

∈

R

such that

⁢

1

+

a

⁢

r

∈

𝔪

.

(1)

Proof.

1

∘

.  Let first

𝔪

be a

maximal ideal

of

R

and

a

∈

R

∖

𝔪

.  Because

𝔪

+

(

a

)

=

R

,  there exist some elements

m

∈

𝔪

and

-

r

∈

R

such that

m

-

a

⁢

r

=

1

.  Consequently,

1

+

a

⁢

r

=

m

∈

𝔪

.

2

∘

.  Assume secondly that the ideal

𝔪

satisfies the condition (1).  Now there must be a maximal ideal

𝔪

′

of

R

such that

𝔪

⊆

𝔪

′

⊂

R

.

Let us make the antithesis that

𝔪

′

∖

𝔪

is non-empty.  Choose an element

a

∈

𝔪

′

∖

𝔪

⊂

R

∖

𝔪

.

By our assumption, we can choose another element

r

of

R

such that

s

=

1

+

a

⁢

r

∈

𝔪

⊂

𝔪

′

.

Then we have

1

=

s

-

a

⁢

r

∈

𝔪

′

+

𝔪

′

=

𝔪

′

which is impossible since with 1 the ideal

𝔪

′

would contain the whole

R

.  Thus the antithesis is wrong and

𝔪

=

𝔪

′

is maximal.

Title

criterion for maximal ideal

Canonical name

CriterionForMaximalIdeal

Date of creation

2013-03-22 19:10:40

Last modified on

2013-03-22 19:10:40

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 16D25

Classification

msc 13A15

Related topic

MaximalIdealIsPrime

Generated on Fri Feb 9 18:37:07 2018 by

LaTeXML
dense ideal

dense ideal

Given a

commutative ring

R

, an ideal/subset

I

⊂

R

is said to be iff its

annihilator

is

{

0

}

, in other words

Ann

⁢

(

I

)

=

{

0

}

We can similarly define and in the case of noncommutative rings.

Title

dense ideal

Canonical name

DenseIdeal

Date of creation

2013-03-22 16:21:23

Last modified on

2013-03-22 16:21:23

Owner

jocaps (12118)

Last modified by

jocaps (12118)

Numerical id

13

Author

jocaps (12118)

Entry type

Definition

Classification

msc 16D25

Defines

dense subset of a ring

Defines

dense subset

Defines

right dense

Defines

left dense

Generated on Fri Feb 9 18:37:09 2018 by

LaTeXML
ideal generated by a subset of a ring

ideal generated by a subset of a ring

Let

X

be a subset of a ring

R

. Let

S

=

{

I

k

}

be the collection of all

left ideals

of

R

that contain

X

(note that the set is nonempty since

X

⊂

R

and

R

is an ideal in itself). The

intersection

I

=

⋂

I

k

∈

S

I

k

is called the

left ideal generated by

X

, and is denoted by

(

X

)

. We say that

X

generates

I

as an ideal.

The definition is symmetrical for

right ideals

.

Alternatively, we can constructively form the set of elements that constitutes this ideal: The left ideal

(

X

)

consists of finite

R

-

linear combinations

of

elements of

X

:

(

X

)

=

{

∑

λ

(

r

λ

⁢

a

λ

+

n

λ

⁢

a

λ

)

∣

a

λ

∈

X

,

r

λ

∈

R

,

n

λ

∈

ℤ

}

.

Title

ideal generated by a subset of a ring

Canonical name

IdealGeneratedByASubsetOfARing

Date of creation

2013-03-22 14:39:04

Last modified on

2013-03-22 14:39:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16D25

Related topic

GeneratorsOfInverseIdeal

Related topic

PrimeIdealsByKrullArePrimeIdeals

Defines

ideal generated by

Defines

left ideal generated by

Defines

right ideal generated by

Defines

generate as an ideal

Defines

generates as an ideal

Defines

generates

Generated on Fri Feb 9 18:37:13 2018 by

LaTeXML
ideal multiplication laws

ideal multiplication laws

The

multiplication

(

http://planetmath.org/ProductOfIdeals

) of the (two-sided) ideals of any ring

R

has following

properties

:

1.

(

0

)

⁢

𝔞

=

𝔞

⁢

(

0

)

=

(

0

)

2.

(

𝔞

⁢

𝔟

)

⁢

𝔠

=

𝔞

⁢

(

𝔟

⁢

𝔠

)

3.

𝔞

⁢

(

𝔟

+

𝔠

)

=

𝔞

⁢

𝔟

+

𝔞

⁢

𝔠

,

(

𝔞

+

𝔟

)

⁢

𝔠

=

𝔞

⁢

𝔠

+

𝔟

⁢

𝔠

4.

If

R

has a unity, then

R

⁢

𝔞

=

𝔞

⁢

R

=

𝔞

5.

If

R

is

commutative

, then

𝔞

⁢

𝔟

=

𝔟

⁢

𝔞

6.

𝔞

⁢

𝔟

⊆

𝔞

∩

𝔟

7.

𝔞

⁢

(

𝔟

∩

𝔠

)

⊆

𝔞

⁢

𝔟

∩

𝔞

⁢

𝔠

8.

𝔞

⊆

𝔟

⇒

𝔞

⁢

𝔠

⊆

𝔟

⁢

𝔠

Remark.

The properties 1, 2, 3, 4 together with the properties

(

𝔞

+

𝔟

)

+

𝔠

=

𝔞

+

(

𝔟

+

𝔠

)

,

𝔞

+

𝔟

=

𝔟

+

𝔞

,

𝔞

+

(

0

)

=

𝔞

of the ideal

addition

make the set

A

of all ideals of

R

to a semiring

(

A

,

+

,

⋅

)

.  It is not a ring, since no non-zero ideal of

R

has the

additive inverse

(

http://planetmath.org/Ring

).

References

1

M. Larsen & P. McCarthy

:

Multiplicative

theory of ideals

.  Academic Press, New York (1971).

Title

ideal multiplication laws

Canonical

name

IdealMultiplicationLaws

Date of creation

2014-05-11 17:05:52

Last modified on

2014-05-11 17:05:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 16D25

Synonym

laws of ideal

product

Related topic

DivisibilityInRings

Related topic

ProductOfLeftAndRightIdeal

Related topic

InvertibilityOfRegularlyGeneratedIdeal

Generated on Fri Feb 9 18:37:16 2018 by

LaTeXML