aarjaneiro/planetmath huggingface dataset:
condition on a near ring to be a ring

condition on a near ring to be a ring

Every ring is a

near-ring

. The

converse

is true only when additional conditions are imposed on the near-ring.

Theorem 1

.

Let

(

R

,

+

,

⋅

)

be a near ring with a

multiplicative identity

1

such that the

⋅

also

left distributes over

+

; that is,

c

⋅

(

a

+

b

)

=

c

⋅

a

+

c

⋅

b

. Then

R

is a ring.

In short, a distributive near-ring with

1

is a ring.

Before proving this, let us list and prove some general facts about a near ring:

1.

Every near ring has a unique

additive identity

: if both

0

and

0

′

are additive identities, then

0

=

0

+

0

′

=

0

′

.

2.

Every

element

in a near ring has a unique

additive inverse

. The additive inverse of

a

is denoted by

-

a

.

Proof.

If

b

and

c

are additive inverses of

a

, then

b

+

a

=

0

=

a

+

c

and

b

=

b

+

0

=

b

+

(

a

+

c

)

=

(

b

+

a

)

+

c

=

0

+

c

=

c

. ∎

3.

-

(

-

a

)

=

a

, since

a

is the (unique) additive inverse of

-

a

.

4.

There is no ambiguity in defining “

subtraction

”

-

on a near ring

R

by

a

-

b

:=

a

+

(

-

b

)

.

5.

a

-

b

=

0

iff

a

=

b

, which is just the

combination

of the above three facts.

6.

If a near ring has a multiplicative identity, then it is unique. The proof is identical to the one given for the first Fact.

7.

If a near ring has a multiplicative identity

1

, then

(

-

1

)

⁢

a

=

-

a

.

Proof.

a

+

(

-

1

)

⁢

a

=

1

⁢

a

+

(

-

1

)

⁢

a

=

(

1

+

(

-

1

)

)

⁢

a

=

0

⁢

a

=

0

. Therefore

(

-

1

)

⁢

a

=

-

a

since

a

has a unique additive inverse. ∎

We are now in the position to prove the

theorem

.

Proof.

Set

r

=

a

+

b

and

s

=

b

+

a

. Then

r

-

s

=

r

-

(

b

+

a

)

substitution

=

r

+

(

-

1

)

⁢

(

b

+

a

)

by Fact

7

above

=

r

+

(

(

-

1

)

⁢

b

+

(

-

1

)

⁢

a

)

by left distributivity

=

r

+

(

-

b

+

(

-

a

)

)

by Fact

7

above

=

(

a

+

b

)

+

(

-

b

+

(

-

a

)

)

substitution

=

(

(

a

+

b

)

+

(

-

b

)

)

+

(

-

a

)

additive associativity

=

(

a

+

(

b

+

(

-

b

)

)

+

(

-

a

)

additive associativity

=

(

a

+

0

)

+

(

-

a

)

-

b

⁢

is the additive inverse of

⁢

b

=

a

+

(

-

a

)

0

⁢

is the additive identity

=

0

same reason as above

Therefore,

a

+

b

=

r

=

s

=

b

+

a

by Fact

5

above.
∎

Title

condition on a near ring to be a ring

Canonical name

ConditionOnANearRingToBeARing

Date of creation

2013-03-22 17:19:54

Last modified on

2013-03-22 17:19:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 20-00

Classification

msc 16-00

Classification

msc 13-00

Related topic

UnitalRing

Generated on Fri Feb 9 19:15:29 2018 by

LaTeXML
correspondence of normal subgroups and group congruences

correspondence of normal subgroups and group congruences

We start with a definition.

Definition 1

.

Let

G

be a group. An

equivalence relation

∼

on

G

is
called a

group

congruence

if it is

compatible

with the
group

structure

, ie. when the following holds

•

∀

a

,

b

,

a

′

,

b

′

∈

G

,

(

a

∼

a

′

and

b

∼

b

′

)

⇒

a

b

∼

a

′

b

′

•

∀

a

,

b

∈

G

,

a

∼

b

⇒

a

-

1

∼

b

-

1

.

So a group congruence is a

http://planetmath.org/node/3403

semigroup

congruence
that additionally preserves the unary operation of taking

inverse

.

It turns out that group congruences correspond to

normal subgroups

:

Theorem 2

.

An equivalence relation

∼

is a group congruence
if and only if there is a normal subgroup such that

∀

a

,

b

∈

G

,

a

∼

b

⟺

a

b

-

1

∈

H

.

Proof.

Let

H

be a

normal

subgroup

of

G

and let

∼

H

be the
equivalence relation

H

defines in

G

. To see that this
equivalence relation is compatible with the

group operation

note
that if

a

′

∼

H

a

and

b

′

∼

H

b

then there are

elements

h

1

and

h

2

of

H

such that

a

′

=

a

⁢

h

1

and

b

′

=

b

⁢

h

2

. Furthermore since

H

is normal in

G

there is an element

h

3

∈

H

such that

h

1

⁢

b

=

b

⁢

h

3

. Then we have

a

′

⁢

b

′

=

a

⁢

h

1

⁢

b

⁢

h

2

=

a

⁢

b

⁢

h

3

⁢

h

2

which gives that

a

′

⁢

b

′

∼

a

⁢

b

.

To prove the

converse

, assume that

∼

is an
equivalence relation compatible with the group operation and let

H

be the

equivalence class

of the

identity

e

. We will prove
that

∼

⁣

=

⁣

∼

H

. We first prove
that

H

is a normal subgroup of

G

. Indeed if

a

∼

e

and

b

∼

e

then by the compatibility we have that

a

⁢

b

1

∼

e

⁢

e

-

1

, that
is

a

⁢

b

-

1

∼

e

; so that

H

is a subgroup of

G

. Now if

g

∈

G

and

h

∈

H

we have

h

∼

e

⇒

g

⁢

h

⁢

g

-

1

∼

g

⁢

e

⁢

g

-

1

⇒

g

⁢

h

⁢

g

-

1

∼

e

⇒

g

⁢

h

⁢

g

-

1

∈

H

.

Therefore

H

is a normal subgroup of

G

. Now consider two
elements

a

and

b

of

G

. To finish the proof observe that for

a

,

b

∈

G

we have

a

∼

H

b

⇒

a

⁢

b

-

1

∈

H

⇒

a

⁢

b

-

1

∼

e

⇒

(

a

⁢

b

-

1

)

⁢

b

∼

e

⁢

b

⇒

a

∼

b

and

a

∼

b

⇒

a

⁢

b

-

1

∼

b

⁢

b

-

1

⇒

a

⁢

b

-

1

∼

e

⇒

a

∼

H

b

.

∎

Title

correspondence of normal subgroups and group congruences

Canonical name

CorrespondenceOfNormalSubgroupsAndGroupCongruences

Date of creation

2013-03-22 15:32:52

Last modified on

2013-03-22 15:32:52

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

7

Author

Dr_Absentius (537)

Entry type

Theorem

Classification

msc 20-00

Defines

group congruence

Generated on Fri Feb 9 19:15:32 2018 by

LaTeXML
cube of a number

cube of a number

The

cube of a number

x

is the

third power

(

http://planetmath.org/GeneralAssociativity

)

x

3

of

x

.  Similarly one may speak of the cube of an

element

x

in any

semigroup

with the

operation

denoted multiplicatively (cf.

general associativity

).

The volume of a cube (i.e.

regular

(

http://planetmath.org/RegularPolyhedron

) hexahedron (

http://planetmath.org/Hexahedron

)) with

a

is

a

3

; hence the name.

The

cube function

x

↦

x

3

from

ℝ

to

ℝ

is

injective

, but not as a

mapping

from

ℂ

to

ℂ

; one has

x

3

=

y

3

always when

x

y

=

-

1

±

i

⁢

3

2

,  the third

root of unity

.

Title

cube of a number

Canonical name

CubeOfANumber

Date of creation

2013-03-22 15:07:13

Last modified on

2013-03-22 15:07:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Definition

Classification

msc 20-00

Synonym

cube

Synonym

third power

Related topic

SquareOfANumber

Related topic

PowerFunction

Related topic

CubeRoot

Related topic

CubanPrime

Related topic

SemicubicalParabola

Defines

cube function

Generated on Fri Feb 9 19:15:34 2018 by

LaTeXML
examples of non-commutative operations

examples of non-commutative operations

A standard example of a

non-commutative operation

is

matrix multiplication

. Consider the following two

integer matrices

:

A

=

(

1

1

0

1

)

,

B

=

(

0

1

0

1

)

If we compute

A

⁢

B

we get

A

⁢

B

=

(

0

2

0

1

)

but if we compute

B

⁢

A

we have

B

⁢

A

=

(

0

1

0

1

)

.

Since

A

⁢

B

≠

B

⁢

A

we conclude that

matrix product

is not

commutative

.

Operations do not necessarily have to operate on numbers. Another classic example is function composition. Let

f

and

g

be

real functions

given by

f

⁢

(

x

)

=

x

2

,

g

⁢

(

x

)

=

2

⁢

x

.

We see that

(

f

∘

g

)

⁢

(

x

)

=

f

⁢

(

g

⁢

(

x

)

)

=

(

2

⁢

x

)

2

=

4

⁢

x

2

,

but

(

g

∘

f

)

⁢

(

x

)

=

g

⁢

(

f

⁢

(

x

)

)

=

2

⁢

(

x

2

)

=

2

⁢

x

2

.

Since we got different

functions

, we conclude that function composition is not commutative.

Title

examples of non-commutative operations

Canonical name

ExamplesOfNoncommutativeOperations

Date of creation

2013-03-22 15:03:04

Last modified on

2013-03-22 15:03:04

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Example

Classification

msc 20-00

Generated on Fri Feb 9 19:15:38 2018 by

LaTeXML
exponentiation

exponentiation

•

In the entry

general associativity

, the notion of the

power

a

n

for elements

a

of a set having an

associative

binary operation

“

⋅

” and for

positive integers

n

as

exponents

(

http://planetmath.org/GeneralPower

) was defined as a generalisation of the operation.  Then the two

power laws

a

m

⋅

a

n

=

a

m

+

n

,

(

a

m

)

n

=

a

m

⁢

n

are .  For the validity of the third well-known power law,

(

a

⋅

b

)

n

=

a

n

⋅

b

n

,

the law of

power of product

, the

commutativity

of the operation is needed.

Example.

In the

symmetric group

S

3

, where the

group operation

is not

commutative

, we get different results from

[

(

123

)

⁢

(

13

)

]

2

=

(

23

)

2

=

(

1

)

and

(

123

)

2

⁢

(

13

)

2

=

(

132

)

⁢

(

1

)

=

(

132

)

(note that in these “

products

”, which

compositions

of mappings, the latter “factor” acts first).

•

Extending the power notion for zero and negative integer exponents requires the existence of

http://planetmath.org/node/10539

neutral and inverse elements (

e

and

a

-

1

):

a

0

:=

e

,

a

-

n

:=

(

a

-

1

)

n

The two first power laws then remain in for all integer exponents, and if the operation is commutative, also the .

When the operation in question is the

multiplication

of real or

complex numbers

, the power notion may be extended for other than integer exponents.

•

One step is to introduce fractional (

http://planetmath.org/FractionalNumber

) exponents by using roots (

http://planetmath.org/NthRoot

); see the fraction power.

•

The following step would be the

irrational

exponents, which are in the

power functions

.  The irrational exponents are possible to introduce by utilizing the

exponential function

and

logarithms

; another way would be to define

a

ϱ

as limit of a

sequence

a

r

1

,

a

r

2

,

…

where the limit of the

rational number

sequence

r

1

,

r

2

,

…

is

ϱ

.  The sequence

a

r

1

,

a

r

2

,

…

may be shown to be a

Cauchy sequence

.

•

The last step were the imaginary (non-real complex) exponents

μ

, when also the base of the power may be other than a positive real number; the one gets the so-called

general power

.

Title

exponentiation

Canonical

name

Exponentiation

Date of creation

2013-03-22 19:08:44

Last modified on

2013-03-22 19:08:44

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 20-00

Related topic

ContinuityOfNaturalPower

Defines

power law

Defines

power of product

Generated on Fri Feb 9 19:15:40 2018 by

LaTeXML
G-Set

G

-Set

If

G

is a group and

X

a set, then

X

is called a left

G

-Set if there exists a mapping

λ

:

G

×

X

→

X

with

λ

⁢

(

g

1

,

λ

⁢

(

g

2

,

x

)

)

=

λ

⁢

(

g

1

⁢

g

2

,

x

)

or shorter with

λ

⁢

(

g

,

x

)

=

g

⁢

x

g

1

⁢

(

g

2

⁢

(

x

)

)

=

(

g

1

⁢

g

2

)

⁢

(

x

)

for all

x

∈

X

and

g

1

,

g

2

∈

G

. And when

G

acts on a set

X

, the set

X

is always a

G

-set.

X

is called a right

G

-Set if there exists a mapping

λ

:

X

×

G

→

X

with

λ

⁢

(

λ

⁢

(

x

,

g

2

)

,

g

1

)

=

λ

⁢

(

x

,

g

2

⁢

g

1

)

Title

G

-Set

Canonical

name

GSet

Date of creation

2013-03-22 17:55:44

Last modified on

2013-03-22 17:55:44

Owner

jwaixs (18148)

Last modified by

jwaixs (18148)

Numerical id

5

Author

jwaixs (18148)

Entry type

Definition

Classification

msc 20-00

Related topic

GroupAction

Generated on Fri Feb 9 19:15:42 2018 by

LaTeXML
general associativity

general associativity

If an

associative

binary operation

of a set

S

is denoted by “

⋅

”, the associative law in

S

is usually expressed as

(

a

⋅

b

)

⋅

c

=

a

⋅

(

b

⋅

c

)

,

or leaving out the dots,

(

a

⁢

b

)

⁢

c

=

a

⁢

(

b

⁢

c

)

.  Thus the common value of both may be denoted as

a

⁢

b

⁢

c

.  With four elements of

S

we can , using only the associativity, as follows:

(

a

⁢

b

)

⁢

(

c

⁢

d

)

=

a

⁢

(

b

⁢

(

c

⁢

d

)

)

=

a

⁢

(

(

b

⁢

c

)

⁢

d

)

=

(

a

⁢

(

b

⁢

c

)

)

⁢

d

=

(

(

a

⁢

b

)

⁢

c

)

⁢

d

So we may denote the common value of those five expressions as

a

⁢

b

⁢

c

⁢

d

.

Theorem

.

The expression formed of

elements

a

1

,

a

2

, …,

a

n

of

S

.  The common value is denoted by

a

1

⁢

a

2

⁢

…

⁢

a

n

.

Note.

The

n

elements can be joined, without changing their , in

(

2

⁢

n

-

2

)

!

n

!

⁢

(

n

-

1

)

!

ways (see the

Catalan numbers

).

The

theorem

is proved by

induction

on

n

.  The cases

n

=

3

and

n

=

4

have been stated above.

Let

n

∈

ℤ

+

.  The expression

a

⁢

a

⁢

…

⁢

a

with

n

equal “factors”

a

may be denoted by

a

n

and called a

power

of

a

.  If the associative

operation

is denoted “additively”, then the “sum”

a

+

a

+

⋯

+

a

of

n

equal elements

a

is denoted by

n

⁢

a

and called a

multiple

of

a

; hence in every ring one may consider powers and multiples. According to whether

n

is an even or an

odd number

, one may speak of

even powers

,

odd powers

,

even multiples

,

odd multiples

.

The following two laws can be proved by induction:

a

m

⋅

a

n

=

a

m

+

n

(

a

m

)

n

=

a

m

⁢

n

In notation:

m

⁢

a

+

n

⁢

a

=

(

m

+

n

)

⁢

a

,

n

⁢

(

m

⁢

a

)

=

(

m

⁢

n

)

⁢

a

Note.

If the set

S

together with its operation is a group, then the notion of multiple

n

⁢

a

resp. power

a

n

can be extended for negative integer and zero values of

n

by means of the

inverse

and

identity elements

.  The above laws remain in .

Title

general associativity

Canonical name

GeneralAssociativity

Date of creation

2013-03-22 14:35:50

Last modified on

2013-03-22 14:35:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

21

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20-00

Related topic

Semigroup

Related topic

EveryRingIsAnIntegerAlgebra

Related topic

InverseFormingInProportionToGroupOperation

Related topic

CosineAtMultiplesOfStraightAngle

Related topic

InfixNotation

Related topic

OperationsOnRelations

Related topic

Difference2

Related topic

FactorsWithMinusSign

Related topic

IdealOfElementsWithFiniteOrder

Related topic

GeneralCommutativity

Related topic

Characteri

Defines

power

Defines

multiple

Defines

even power

Defines

odd power

Defines

even multiple

Defines

odd multiple

Generated on Fri Feb 9 19:15:44 2018 by

LaTeXML
general commutativity

general commutativity

Theorem

.

If the

binary operation

“

⋅

” on the set

S

is

commutative

, then for each

a

1

,

a

2

,

…

,

a

n

in

S

and for each

permutation

π

on

{

1

,

2

,

…

,

n

}

,  one has

∏

i

=

1

n

a

π

⁢

(

i

)

=

∏

i

=

1

n

a

i

.

(1)

Proof.

If

n

=

1

,  we have nothing to prove.
Make the

induction hypothesis

, that (1) is true for

n

=

m

-

1

.  Denote

π

-

1

⁢

(

m

)

=

k

,

i.e.

π

⁢

(

k

)

=

m

.

Then

∏

i

=

1

m

a

π

⁢

(

i

)

=

∏

i

=

1

k

-

1

a

π

⁢

(

i

)

⋅

a

π

⁢

(

k

)

⋅

∏

i

=

1

m

-

k

a

π

⁢

(

k

+

i

)

=

(

∏

i

=

1

k

-

1

a

π

⁢

(

i

)

⋅

∏

i

=

1

m

-

k

a

π

⁢

(

k

+

i

)

)

⋅

a

m

,

where

a

m

has been moved to the end by the induction
hypothesis.  But the

product

in the parenthesis, which
exactly the factors

a

1

,

a

2

,

…

,

a

m

-

1

in a certain , is also by the induction hypothesis equal to

∏

i

=

1

m

-

1

a

i

.  Thus we obtain

∏

i

=

1

m

a

π

⁢

(

i

)

=

∏

i

=

1

m

-

1

a

i

⋅

a

m

=

∏

i

=

1

m

a

i

,

whence (1) is true for

n

=

m

.

Note.

There is mentionned in the Remark of the entry “

http://planetmath.org/node/2148

commutativity” a more general notion of commutativity.

Title

general commutativity

Canonical name

GeneralCommutativity

Date of creation

2014-05-10 21:59:41

Last modified on

2014-05-10 21:59:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20-00

Related topic

CommutativeLanguage

Related topic

GeneralAssociativity

Related topic

AbelianGroup2

Generated on Fri Feb 9 19:15:46 2018 by

LaTeXML
generated subring

generated subring

Definition 1

Let

M

be a nonempty subset of a ring

A

. The

intersection

of all subrings of

A

that include

M

is the smallest subring of

A

that includes

M

. It is called the

subring generated by

M

and is denoted by

⟨

M

⟩

.

The subring generated by

M

is formed by finite sums of

monomials

of the form :

a

1

⁢

a

2

⁢

⋯

⁢

a

n

,

where

⁢

a

1

,

…

,

a

n

∈

M

.

Of particular

interest

is the subring generated by a family of subrings

E

=

{

A

i

|

i

∈

I

}

. It is the ring

R

formed by finite sums of monomials of the form:

a

i

1

⁢

a

i

2

⁢

…

⁢

a

i

n

,

where

⁢

a

i

k

∈

A

i

k

.

If

A

,

B

are rings, the subring generated by

A

∪

B

is also denoted by

A

⁢

B

.

In the case when

A

i

are fields included in a larger field

A

then the set of all

quotients

of elements of

R

( the

quotient field

of

R

) is the composite field

⋁

i

∈

I

A

i

of the family

E

. In other words, it is the subfield generated by

⋃

i

∈

I

A

i

. The notation

⋁

i

∈

I

A

i

comes from the fact that the family of all subfields of a field forms a

complete lattice

.

The of fields is defined only when the respective fields are all included in a larger field.

Title

generated subring

Canonical

name

GeneratedSubring

Date of creation

2013-03-22 16:57:27

Last modified on

2013-03-22 16:57:27

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

9

Author

polarbear (3475)

Entry type

Definition

Classification

msc 20-00

Classification

msc 13-00

Classification

msc 16-00

Related topic

RingAdjunction

Defines

subring generated by

Defines

monomials in rings

Generated on Fri Feb 9 19:15:48 2018 by

LaTeXML
indecomposable group

indecomposable group

By definition, an

indecomposable group

is a nontrivial group that cannot be expressed as the internal

direct product

of two proper

normal subgroups

. A group that is not

indecomposable

is called, predictably enough,

decomposable

.

The analogous

concept

exists in module theory. An

indecomposable module

is a nonzero module that cannot be expressed as the

direct sum

of two nonzero

submodules

.

The following examples are left as exercises for the reader.

1.

Every

simple group

is indecomposable.

2.

If

p

is prime and

n

is any positive integer, then the

additive group

ℤ

/

p

n

⁢

ℤ

is indecomposable. Hence, not every indecomposable group is simple.

3.

The additive groups

ℤ

and

ℚ

are indecomposable, but the additive group

ℝ

is decomposable.

4.

If

m

and

n

are relatively prime integers (and both greater than one), then the additive group

ℤ

/

m

⁢

n

⁢

ℤ

is decomposable.

5.

Every

finitely generated

abelian group

can be expressed as the direct sum of finitely many indecomposable groups. These summands are uniquely determined up to

isomorphism

.

References

.

•

Dummit, D. and R. Foote,

Abstract Algebra

. (2d ed.), New York: John Wiley and Sons, Inc., 1999.

•

Goldhaber, J. and G. Ehrlich,

Algebra

. London: The Macmillan Company, 1970.

•

Hungerford, T.,

Algebra

. New York: Springer, 1974.

Title

indecomposable group

Canonical name

IndecomposableGroup

Date of creation

2013-03-22 15:23:46

Last modified on

2013-03-22 15:23:46

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20-00

Synonym

indecomposable

Related topic

KrullSchmidtTheorem

Defines

decomposable

Defines

indecomposable module

Generated on Fri Feb 9 19:15:50 2018 by

LaTeXML
inverse of inverse in a group

inverse of inverse in a group

Let

(

G

,

*

)

be a group.
We aim to prove that

(

a

-

1

)

-

1

=

a

for every

a

∈

G

.
That is, the

inverse

of the inverse of a group element is the element itself.

By

definition

a

*

a

-

1

=

a

-

1

*

a

=

e

, where

e

is the identity in

G

. Reinterpreting this equation we can read it as saying that

a

is the inverse of

a

-

1

.

In fact, consider

b

=

a

-

1

, the equation can be written

a

*

b

=

b

*

a

=

e

and thus

a

is the inverse of

b

=

a

-

1

.

Title

inverse of inverse in a group

Canonical

name

InverseOfInverseInAGroup

Date of creation

2013-03-22 15:43:36

Last modified on

2013-03-22 15:43:36

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Proof

Classification

msc 20-00

Classification

msc 20A05

Classification

msc 08A99

Related topic

AdditiveInverseOfAnInverseElement

Generated on Fri Feb 9 19:15:51 2018 by

LaTeXML
Klein 4-ring

Klein 4-ring

One of the two smallest

non-commutative

rings is the

Klein 4-ring

(

R

,

+

,

⋅

)

where

(

R

,

+

)

is the

Klein 4-group

{

0

,

a

,

b

,

c

}

with

0

the

neutral element

and the

binary operation

“

⋅

” given by the table

⋅

0

a

b

c

0

0

0

0

0

a

0

a

0

a

b

0

b

0

b

c

0

c

0

c

Note that this ring has two different

right unities

a

and

c

.

The Klein 4-ring has the

subrings

{

0

,

a

}

,

{

0

,

b

}

and

{

0

,

c

}

and the

two-sided ideal

{

0

,

b

}

.

Title

Klein 4-ring

Canonical name

Klein4ring

Date of creation

2015-06-04 16:11:18

Last modified on

2015-06-04 16:11:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 20-00

Classification

msc 16B99

Synonym

Klein’s four-ring

Synonym

Klein four-ring

Related topic

Klein4Group

Related topic

InversesInRings

Related topic

NonCommutativeRingsOfOrderFour

Related topic

GroupsInField

Related topic

Subcommutative

Generated on Fri Feb 9 19:15:53 2018 by

LaTeXML
law of signs under multiplication in a ring

law of signs under multiplication in a ring

Lemma 1

.

Let

R

be a

ring with unity

, which we denote by

1

. For all

x

,

y

∈

R

:

(

-

x

)

⋅

(

-

y

)

=

x

⋅

y

where

-

x

denotes the

additive inverse

of

x

in

R

.

Proof.

Here we use the fact

(

-

1

)

⋅

a

=

-

a

for all

a

∈

R

. First, we see that:

(

-

1

)

⋅

(

-

1

)

⋅

a

=

(

-

1

)

⋅

(

(

-

1

)

⋅

a

)

=

(

-

1

)

⋅

(

-

a

)

=

a

since, clearly, the additive inverse of

-

a

is

a

itself.

Hence:

(

-

x

)

⋅

(

-

y

)

=

(

-

1

)

⋅

x

⋅

(

-

1

)

⋅

y

=

(

-

1

)

⋅

(

-

1

)

⋅

x

⋅

y

=

x

⋅

y

where we have used several times the

associativity

of

⋅

and the fact that

(

-

1

)

⋅

x

=

x

⋅

(

-

1

)

=

-

x

.
∎

Title

law of signs under

multiplication

in a ring

Canonical

name

LawOfSignsUnderMultiplicationInARing

Date of creation

2013-03-22 14:14:03

Last modified on

2013-03-22 14:14:03

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Derivation

Classification

msc 20-00

Classification

msc 16-00

Classification

msc 13-00

Synonym

(

-

x

)

⋅

(

-

y

)

=

x

⋅

y

Related topic

Ring

Generated on Fri Feb 9 19:15:55 2018 by

LaTeXML