aarjaneiro/planetmath huggingface dataset:
category of H * -algebras

category of H

*

-algebras

Definition 0.1

.

The

category

of H

*

-

algebras

is defined as the category whose objects
are H

*

-algebras (

http://planetmath.org/HAlgebra

) and whose

morphisms

are *-

homomorphisms

between H

*

-algebras
that commute with the antilinear

involution

*

:

𝔸

H

→

𝔸

H

.

Remark 0.1

.

The construction of H

*

-algebras is sometimes called ‘

groupoidification

’

Title

category of H

*

-algebras

Canonical name

CategoryOfHalgebras

Date of creation

2013-03-22 18:26:45

Last modified on

2013-03-22 18:26:45

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

10

Author

bci1 (20947)

Entry type

Definition

Classification

msc 20G42

Classification

msc 46N50

Classification

msc 18-00

Classification

msc 81R05

Classification

msc 81R15

Synonym

groupoidification

Related topic

HAlgebra

Related topic

IndexOfCategories

Defines

groupoidification

Generated on Fri Feb 9 19:31:35 2018 by

LaTeXML
H * -algebra

H * -algebra

0.1

H *-algebra

An

H

*

-

algebra

is defined as a

Hilbert space

𝔸

H

equipped with an associative unital algebra

structure

and an antilinear

involution

:

*

𝔸

H

→

𝔸

H

which is

compatible

with taking the

adjoint

of the

operators

on the Hilbert space for the left and right

multiplication

of

𝔸

H

with itself (ref.

[

1

]

).

References

1

Baez, J. 2007.

http://golem.ph.utexas.edu/category/2006/10/categorified_gelfandnaimark_th.html

Groupoidification

.
(Preprint)

Title

H * -algebra

Canonical name

Halgebra

Date of creation

2013-03-22 18:26:32

Last modified on

2013-03-22 18:26:32

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

7

Author

bci1 (20947)

Entry type

Definition

Classification

msc 20G42

Classification

msc 81R05

Classification

msc 81R15

Classification

msc 46N50

Related topic

HilbertSpace

Related topic

QuantumSpaceTimes

Related topic

VonNeumannAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

CategoryOfHAlgebras

Generated on Fri Feb 9 19:31:37 2018 by

LaTeXML
unimodular matrix

unimodular matrix

An

n

×

n

square matrix

over a field is

unimodular

if its

determinant

is 1. The set of all

n

×

n

unimodular matrices

forms a group under the usual

matrix multiplication

. This group is known as the

special linear group

. Any of its

subgroup

is simply called a

unimodular group

. Furthermore, unimodularity is preserved under

similarity transformations

: if

S

any

n

×

n

invertible matrix

and

U

is unimodular, then

S

-

1

⁢

U

⁢

S

is unimodular. In view of the last statement, the special linear group is a

normal subgroup

of the group of all invertible matrices, known as the

general linear group

.

A

linear transformation

T

over an

n

-dimensional

vector space

V

(over a field

F

) is

unimodular

if it can be represented by a unimodular matrix.

The concept of the unimodularity of a square matrix over a field can be readily extended to that of a square matrix over a commutative ring. Unimodularity in square matrices can even be extended to arbitrary

finite-dimensional

matrices: suppose

R

is a commutative ring with 1, and

M

is an

m

×

n

matrix over

R

(entries are elements of

R

) with

m

≤

n

. Then

M

is said to be

unimodular

if it can be “completed” to a

n

×

n

square unimodular matrix

N

over

R

. By completion of

M

to

N

we mean that

m

of the

n

rows in

N

are exactly the rows of

M

. Of course, the operation of completion from a matrix to a square matrix can be done via columns too.

Let

M

is an

m

×

n

matrix and

v

is any row of

M

. If

M

is unimodular, then

v

is unimodular viewed as a

1

×

n

matrix. A

1

×

n

unimodular matrix is called a

unimodular row

, or a

unimodular vector

. A

n

×

1

unimodular column

can be defined via a

similar

procedure. Let

v

=

(

v

1

,

…

,

v

n

)

be a

1

×

n

matrix over

R

. Then the unimodularity of

v

means that

v

1

⁢

R

+

⋯

+

v

n

⁢

R

=

R

.

To see this, let

U

be a completion of

v

with

det

⁡

(

U

)

=

1

. Since

det

is a

multilinear

operator over the rows (or columns) of

U

, we see that

1

=

det

⁡

(

U

)

=

v

1

⁢

r

1

+

⋯

+

v

n

⁢

r

n

.

Title

unimodular matrix

Canonical name

UnimodularMatrix

Date of creation

2013-03-22 14:57:50

Last modified on

2013-03-22 14:57:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20H05

Classification

msc 15A04

Classification

msc 15A09

Related topic

SpecialLinearGroup

Defines

unimodular linear transformation

Defines

unimodular row

Defines

unimodular column

Defines

unimodular group

Defines

unimodular vector

Generated on Fri Feb 9 19:31:39 2018 by

LaTeXML
monomial matrix

monomial matrix

Let

A

be a matrix with entries in a field

K

. If in every

http://planetmath.org/node/2464

row and every

http://planetmath.org/node/2464

column of

A

there is exactly one nonzero entry, then

A

is a

monomial matrix

.

Obviously, a monomial matrix is a

square matrix

and there exists a
rearrangement of and such that the result is a

diagonal
matrix

.

The

n

×

n

monomial matrices form a group under

matrix
multiplication

. This group contains the

n

×

n

permutation
matrices

as a

subgroup

. A monomial matrix is

invertible

but, unlike a
permutation matrix, not necessarily

http://planetmath.org/node/1176

orthogonal

. The only exception is
when

K

=

𝔽

2

(the finite field with

2

elements

), where the

n

×

n

monomial matrices and the

n

×

n

permutation matrices
coincide.

Title

monomial matrix

Canonical name

MonomialMatrix

Date of creation

2013-03-22 15:15:51

Last modified on

2013-03-22 15:15:51

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

5

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 20H20

Classification

msc 15A30

Related topic

PermutationMatrix

Generated on Fri Feb 9 19:31:42 2018 by

LaTeXML
factor system

factor system

Using the notation in the entry

Group Cohomology

, a 2-cocycle or

factor system

is a map

ϕ

:

G

×

G

→

M

such that for all

α

,

β

,

γ

∈

G

, we have

α

⁢

ϕ

⁢

(

β

,

γ

)

-

ϕ

⁢

(

α

⁢

β

,

γ

)

+

ϕ

⁢

(

α

,

β

⁢

γ

)

-

ϕ

⁢

(

β

,

γ

)

=

0

.

Factor systems play a role in determing classes of

group extensions

of

G

by

M

in the case where

M

is taken to be an

abelian

normal subgroup

of

G

.

Title

factor system

Canonical name

FactorSystem

Date of creation

2013-03-22 14:23:28

Last modified on

2013-03-22 14:23:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20J06

Generated on Fri Feb 9 19:31:47 2018 by

LaTeXML
stronger Hilbert theorem 90

stronger Hilbert theorem 90

Let

K

be a field and let

K

¯

be an

algebraic closure

of

K

. By

K

¯

+

we denote the

abelian group

(

K

¯

,

+

)

and
similarly

K

¯

∗

=

(

K

¯

,

∗

)

(here the operation is

multiplication

). Also we let

G

K

¯

/

K

=

Gal

⁡

(

K

¯

/

K

)

be the

absolute
Galois group

of

K

.

Theorem 1

(Hilbert 90).

Let

K

be a field.

1.

H

1

⁢

(

G

K

¯

/

K

,

K

¯

+

)

=

0

2.

H

1

⁢

(

G

K

¯

/

K

,

K

¯

∗

)

=

0

3.

If

char

⁡

(

K

)

, the

characteristic

of

K

, does
not divide

m

(or

char

⁡

(

K

)

=

0

) then

H

1

⁢

(

G

K

¯

/

K

,

μ

m

)

≅

K

∗

/

K

∗

m

where

μ

m

denotes the set of all

m

t

⁢

h

-

roots of unity

.

References

1

J.P. Serre,

Galois Cohomology

,
Springer-Verlag, New York.

2

J.P. Serre,

Local Fields

,
Springer-Verlag, New York.

Title

stronger Hilbert theorem 90

Canonical

name

StrongerHilbertTheorem90

Date of creation

2013-03-22 13:50:27

Last modified on

2013-03-22 13:50:27

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20J06

Synonym

Hilbert 90

Related topic

HilbertTheorem90

Generated on Fri Feb 9 19:31:53 2018 by

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prime residue class

prime residue class

Let

m

be a

positive integer

. There are

m

residue classes

a

+

m

⁢

ℤ

modulo

m

.  Such of them which have

gcd

⁡

(

a

,

m

)

=

1

,

are called the

prime residue classes

or

prime classes modulo

m

, and they form an

Abelian group

with respect to the

multiplication

(

a

+

m

⁢

ℤ

)

⋅

(

b

+

m

⁢

ℤ

)

:=

a

⁢

b

+

m

⁢

ℤ

.

This group is called the

residue class group modulo

m

. Its order is

φ

⁢

(

m

)

, where

φ

means Euler’s

totient function

. For example, the prime classes modulo 8 (i.e.

1

+

8

⁢

ℤ

,

3

+

8

⁢

ℤ

,

5

+

8

⁢

ℤ

,

7

+

8

⁢

ℤ

) form a group

isomorphic

to the

Klein 4-group

.

The prime classes are the units of the residue class ring

ℤ

/

m

⁢

ℤ

=

ℤ

m

consisting of all residue classes modulo

m

.

Analogically, in the ring

R

of

integers

(

http://planetmath.org/ExamplesOfRingOfIntegersOfANumberField

) of any

algebraic number field

, there are the residue classes and the prime residue classes modulo an ideal

𝔞

of

R

. The number of all residue classes is

N

⁢

(

𝔞

)

and the number of the prime classes is also denoted by

φ

⁢

(

𝔞

)

.  It may be proved that

φ

⁢

(

𝔞

)

=

N

⁢

(

𝔞

)

⁢

∏

𝔭

|

𝔞

(

1

-

1

N

⁢

(

𝔭

)

)

;

N

is the

absolute norm of ideal

and

𝔭

runs all distinct

prime ideals

dividing

𝔞

(cf. the first

formula

in the entry “

Euler phi function

(

http://planetmath.org/EulerPhiFunction

)”).  Moreover, one has the result

α

φ

⁢

(

𝔞

)

≡

1

(

mod

𝔞

)

for

(

(

a

)

,

𝔞

)

=

(

1

)

,  generalising the Euler–

Fermat theorem

(

http://planetmath.org/EulerFermatTheorem

).

Title

prime residue class

Canonical name

PrimeResidueClass

Date of creation

2013-03-22 15:43:12

Last modified on

2013-03-22 15:43:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 20K01

Classification

msc 13M99

Classification

msc 11A07

Synonym

prime class

Related topic

MultiplicativeOrderOfAnIntegerModuloM

Related topic

NonZeroDivisorsOfFiniteRing

Related topic

GroupOfUnits

Related topic

PrimitiveRoot

Related topic

ResidueSystems

Related topic

Klein4Group

Related topic

EulerPhifunction

Related topic

SummatoryFunctionOfArithmeticFunction

Defines

residue class group

Generated on Fri Feb 9 19:31:55 2018 by

LaTeXML
Schinzel’s theorem

Schinzel’s theorem

Definition 1

.

Let

A

and

B

be

integers

such that

(

A

,

B

)

=

1

with

A

⁢

B

≠

±

1

. A prime

p

is called a

primitive

divisor

of

A

n

-

B

n

if

p

divides

A

n

-

B

n

but

A

m

-

B

m

is not

divisible

by

p

for all

positive integers

m

that are less than

n

.

Or, more generally:

Definition 2

.

Let

A

and

B

be

algebraic integers

in a

number field

K

such that

(

A

,

B

)

=

1

and

A

/

B

is not a

root of unity

. A

prime ideal

℘

of

K

is called a

primitive divisor

of

A

n

-

B

n

if

℘

|

A

n

-

B

n

but

℘

∤

A

m

-

B

m

for all positive integers

m

that are less than

n

.

The following

theorem

is due to A. Schinzel (see

[

1

]

):

Theorem

.

Let

A

and

B

be as before. There is an effectively

computable

constant

n

0

, depending only on the degree of the

algebraic number

A

/

B

, such that

A

n

-

B

n

has a primitive divisor for all

n

>

n

0

.

By putting

B

=

1

we obtain the following corollary:

Corollary

.

Let

A

≠

0

,

±

1

be an integer. There exists a number

n

0

such that

A

n

-

1

has a primitive divisor for all

n

>

n

0

. In particular, for all but finitely many integers

n

, there is a prime

p

such that the

multiplicative order

of

A

modulo

p

is exactly

n

.

References

1

A. Schinzel,

Primitive divisors of the expression

A

n

-

B

n

in

algebraic number fields

.

Collection

of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. J. Reine Angew. Math. 268/269 (1974), 27–33.

Title

Schinzel’s theorem

Canonical name

SchinzelsTheorem

Date of creation

2013-03-22 12:03:42

Last modified on

2013-03-22 12:03:42

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20K01

Generated on Fri Feb 9 19:31:57 2018 by

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torsion

torsion

The

torsion

of a group

G

is the set

Tor

(

G

)

=

{

g

∈

G

:

g

n

=

e

⁢

for some

n

∈

ℕ

}

.

A group is said to be

torsion-free

if

Tor

(

G

)

=

{

e

}

,
i.e. the torsion consists only of the

identity element

.

If

G

is

abelian

(or, more generally,

locally nilpotent

) then

Tor

(

G

)

is a

subgroup

(the

torsion subgroup

) of

G

.
Whenever

Tor

(

G

)

is a subgroup of

G

, then it is

fully invariant

and

G

/

Tor

(

G

)

is torsion-free.

Example 1

(Torsion of a finite group)

For any

finite group

G

,

Tor

(

G

)

=

G

.

Example 2

(Torsion of the circle group)

The torsion of the circle group

R

/

Z

is

Tor

(

R

/

Z

)

=

Q

/

Z

.

Title

torsion

Canonical name

Torsion

Date of creation

2013-03-22 13:21:38

Last modified on

2013-03-22 13:21:38

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

8

Author

mhale (572)

Entry type

Definition

Classification

msc 20K10

Synonym

group torsion

Related topic

PeriodicGroup

Defines

torsion-free

Defines

torsion group

Defines

torsion subgroup

Defines

torsion free

Generated on Fri Feb 9 19:31:59 2018 by

LaTeXML
Baer-Specker group

Baer-Specker group

Let

A

be a non-empty set, and

G

an

abelian group

. The set

K

of all

functions

from

A

to

G

is an abelian group, with

addition

defined elementwise by

(

f

+

g

)

⁢

(

x

)

=

f

⁢

(

x

)

+

g

⁢

(

x

)

. The

zero element

is
the function that sends all elements of

A

into

0

of

G

, and the

negative

of an

element

f

is a function defined by

(

-

f

)

⁢

(

x

)

=

-

(

f

⁢

(

x

)

)

.

When

A

=

ℕ

, the set of natural numbers, and

G

=

ℤ

,

K

as defined above is called the

Baer-Specker group

. Any
element of

K

, being a function from

ℕ

to

ℤ

,
can be expressed as an

infinite

sequence

(

x

1

,

x

2

,

…

,

x

n

,

…

)

, and the elementwise addition on

K

can
be realized as componentwise addition on the sequences:

(

x

1

,

x

2

,

…

,

x

n

,

…

)

+

(

y

1

,

y

2

,

…

,

y

n

,

…

)

=

(

x

1

+

y

1

,

x

2

+

y

2

,

…

,

x

n

+

y

n

,

…

)

.

An alternative

characterization

of the Baer-Specker group

K

is that it can be
viewed as the

countably infinite

direct product

of copies of

ℤ

:

K

=

ℤ

ℕ

≅

ℤ

ℵ

0

=

∏

ℵ

0

ℤ

.

The Baer-Specker group is an important example of a torsion-free
abelian group whose rank is infinite. It is not a

free abelian
group

, but any of its

countable

subgroup

is free (

abelian

).

References

1

P. A. Griffith,

Infinite Abelian Group Theory

, The University of Chicago Press (1970)

Title

Baer-Specker group

Canonical

name

BaerSpeckerGroup

Date of creation

2013-03-22 15:29:18

Last modified on

2013-03-22 15:29:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20K20

Synonym

Specker group

Generated on Fri Feb 9 19:32:02 2018 by

LaTeXML
external direct product of groups

external direct product of groups

The

external direct product

G

×

H

of two groups

G

and

H

is defined to be the set of

ordered pairs

(

g

,

h

)

, with

g

∈

G

and

h

∈

H

. The

group operation

is defined by

(

g

,

h

)

⁢

(

g

′

,

h

′

)

=

(

g

⁢

g

′

,

h

⁢

h

′

)

It can be shown that

G

×

H

obeys the group axioms. More generally, we can define the external direct product of

n

groups, in the obvious way. Let

G

=

G

1

×

…

×

G

n

be the set of all ordered n-tuples

{

(

g

1

,

g

2

⁢

…

,

g

n

)

∣

g

i

∈

G

i

}

and define the group operation by componentwise

multiplication

as before.

Title

external direct product of groups

Canonical name

ExternalDirectProductOfGroups

Date of creation

2013-03-22 12:23:17

Last modified on

2013-03-22 12:23:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20K25

Synonym

direct product

Related topic

CategoricalDirectProduct

Related topic

DirectProductAndRestrictedDirectProductOfGroups

Generated on Fri Feb 9 19:32:04 2018 by

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proof of fundamental theorem of finitely generated abelian groups

proof of fundamental theorem of finitely generated abelian groups

Every

finitely generated

abelian group

A

is a

direct sum

of its

cyclic subgroups

, i.e.

A

=

C

m

1

⊕

C

m

2

⊕

…

⊕

C

m

k

⊕

ℤ

⊕

…

⊕

ℤ

,

where

1

⁢

<

m

1

∣

⁢

m

2

⁢

∣

…

∣

⁢

m

k

. The numbers

m

i

are uniquely determined as well as the number of

ℤ

’s, which is the rank of an abelian group.

Proof. Let

G

be an abelian group with

n

generators

. Then for a

free group

F

n

,

G

is

isomorphic

to the

quotient group

F

n

/

A

. Now

F

n

and

A

contain a basis

f

1

,

…

,

f

n

and

a

1

,

…

,

a

k

satisfying

a

i

=

m

i

⁢

f

i

for all

1

≤

i

≤

k

. As

G

≅

F

n

/

A

, it suffices to show that

F

n

/

A

is a direct sum of its cyclic subgroups

⟨

f

1

+

A

⟩

.

It is clear that

F

n

/

A

is generated by its

subgroups

⟨

f

i

+

A

⟩

. Assume that the

zero element

of

F

n

/

A

can be written as a form

A

=

l

1

⁢

f

1

+

…

+

l

n

⁢

f

n

+

A

. It follows that

l

1

⁢

f

1

+

…

+

l

n

⁢

f

n

=

a

∈

A

. As we write

a

as a

linear combination

of that basis and using

a

i

=

m

i

⁢

f

i

we get the

equations

l

1

⁢

f

1

+

…

+

l

n

⁢

f

n

=

s

1

⁢

a

1

+

…

⁢

s

k

⁢

a

k

=

s

1

⁢

m

1

⁢

f

1

+

…

+

s

k

⁢

m

k

⁢

f

k

.

As every element can be represented uniquely as a linear combination of its free generators

f

1

, we have

l

i

=

s

i

⁢

m

i

for every

1

≤

i

≤

k

and

l

j

=

0

for every

k

<

j

≤

n

.

This means that every element

l

i

⁢

f

i

belongs to

A

, so

l

i

⁢

f

i

+

A

=

A

. Therefore the zero element has a unique

representation

as a sum of the elements of the subgroup

⟨

f

i

+

A

⟩

.

References

1

P. Paajanen:

Ryhmäteoria

.  Lecture notes, Helsinki university, Finland (fall 2008)

Title

proof of fundamental theorem of finitely generated abelian groups

Canonical name

ProofOfFundamentalTheoremOfFinitelyGeneratedAbelianGroups

Date of creation

2013-03-22 18:24:58

Last modified on

2013-03-22 18:24:58

Owner

puuhikki (9774)

Last modified by

puuhikki (9774)

Numerical id

7

Author

puuhikki (9774)

Entry type

Proof

Classification

msc 20K25

Generated on Fri Feb 9 19:32:07 2018 by

LaTeXML
pure subgroup

pure subgroup

Definition

. A

pure subgroup

H

of an

abelian group

G

is

1.

a

subgroup

of

G

, such that

2.

H

∩

m

⁢

G

=

m

⁢

H

for all

m

∈

ℤ

.

The second condition says that for any

h

∈

H

such that

h

=

m

⁢

a

for
some integer

m

and some

a

∈

G

, then there exists

b

∈

H

such
that

h

=

m

⁢

b

. In other words, if

h

is divisible in

G

by an
integer, then it is divisible in

H

by that same integer. Purity in
abelian groups is a relative notion, and we denote

H

<

p

G

to mean
that

H

is a pure subgroup of

G

.

Examples

. All groups mentioned below are abelian groups.

1.

For any group, two trivial examples of pure subgroups are the trivial
subgroup and the group itself.

2.

Any divisible subgroup (

http://planetmath.org/DivisibleGroup

) or any

direct summand

of a group is pure.

3.

The

torsion subgroup

(= the subgroup of all

torsion elements

) of any group is pure.

4.

If

K

<

p

H

,

H

<

p

G

, then

K

<

p

G

.

5.

If

H

=

⋃

i

=

1

∞

H

i

with

H

i

≤

H

i

+

1

and

H

i

<

p

G

, then

H

<

p

G

.

6.

In

Z

n

2

,

⟨

n

⟩

is an example of a subgroup that is not
pure.

7.

In general,

⟨

m

⟩

<

p

Z

n

if

gcd

⁡

(

s

,

t

)

=

1

,
where

s

=

gcd

⁡

(

m

,

n

)

and

t

=

n

/

s

.

Remark

. This definition can be generalized to modules over

commutative rings

.

Definition

. Let

R

be a commutative ring and

ℰ

:

0

→

A

→

B

→

C

→

0

a

short exact sequence

of

R

-modules. Then

ℰ

is said to be

pure

if it remains exact after
tensoring with any

R

-module. In other words, if

D

is any

R

-module, then

D

⊗

ℰ

:

0

→

D

⊗

A

→

D

⊗

B

→

D

⊗

C

→

0

,

is exact.

Definition

. Let

N

be a

submodule

of

M

over a ring

R

.
Then

N

is said to be a

pure submodule

of

M

if the

exact
sequence

0

→

N

→

M

→

M

/

N

→

0

is a pure exact sequence.

From this definition, it is clear that

H

is a pure subgroup of

G

iff

H

is a pure

ℤ

-submodule of

G

.

Remark

.

N

is a pure submodule of

M

over

R

iff
whenever a finite sum

∑

r

i

⁢

m

i

=

n

∈

N

,

where

m

i

∈

M

and

r

i

∈

R

implies that

n

=

∑

r

i

⁢

n

i

for some

n

i

∈

N

. As a
result, if

I

is an ideal of

R

, then the purity of

N

in

M

means
that

N

∩

I

⁢

M

=

I

⁢

N

, which is a generalization of the second
condition in the definition of a pure subgroup above.

Title

pure subgroup

Canonical name

PureSubgroup

Date of creation

2013-03-22 14:57:47

Last modified on

2013-03-22 14:57:47

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20K27

Classification

msc 13C13

Defines

pure submodule

Defines

pure exact sequence

Generated on Fri Feb 9 19:32:13 2018 by

LaTeXML
divisible closure

divisible closure

Let

G

be a torsion-free abelian group. We will say that

D

is the

divisible closure

of

G

if

D

is the only divisible

torsion-free

group between

G

and

D

, or equivalently, if for each

x

∈

D

there exists a non-negative integer

n

such that

n

⁢

x

∈

G

.

Title

divisible closure

Canonical name

DivisibleClosure

Date of creation

2013-03-22 15:58:36

Last modified on

2013-03-22 15:58:36

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

5

Author

polarbear (3475)

Entry type

Definition

Classification

msc 20K35

Defines

divisible closure of a group

Generated on Fri Feb 9 19:32:21 2018 by

LaTeXML