examples of rings

examples of rings

Rings in this article are assumed to have a

commutative

addition

with

negatives

and an

associative

multiplication

. However, it
is not generally assumed that all rings included here are unital.

Examples of commutative rings

1.

the

zero ring

,

2.

the

ring of integers

ℤ

,

3.

the ring of

even integers

2

⁢

ℤ

(a ring without

identity

), or more generally,

n

⁢

ℤ

for any

integer

n

,

4.

the integers modulo

n

(

http://planetmath.org/MathbbZ_n

),

ℤ

/

n

⁢

ℤ

,

5.

the ring of integers

𝒪

K

of a

number field

K

,

6.

the

p

-

integral

rational numbers

(

http://planetmath.org/PAdicValuation

) (where

p

is a

prime number

),

7.

other rings of rational numbers

8.

the

p

-adic integers (

http://planetmath.org/PAdicIntegers

)

ℤ

p

and the

p

-adic numbers

ℚ

p

,

9.

the rational numbers

ℚ

,

10.

the

real numbers

ℝ

,

11.

rings and

fields of algebraic numbers

,

12.

the

complex numbers

ℂ

,

13.

The set

2

A

of all subsets of a set

A

is a ring. The addition is the

symmetric difference

“

△

” and the multiplication the set

operation

intersection

“

∩

”. Its

additive identity

is the

empty set

∅

, and its

multiplicative identity

is the set

A

. This is an example of a

Boolean ring

.

Examples of non-commutative rings

1.

the

quaternions

,

ℍ

, also known as the Hamiltonions. This is a finite dimensional

division ring

over the real numbers, but

noncommutative

.

2.

the set of

square matrices

M

n

⁢

(

R

)

, with

n

>

1

,

3.

the set of

triangular matrices

(upper or lower, but not both in the same set),

4.

strict triangular matrices (

http://planetmath.org/StrictUpperTriangularMatrix

) (same condition as above),

5.

Klein 4-ring

,

6.

Let

A

be an

abelian group

. Then the set of

group endomorphisms

f

:

A

→

A

forms a ring

End

⁡

A

,
with addition defined elementwise (

(

f

+

g

)

⁢

(

a

)

=

f

⁢

(

a

)

+

g

⁢

(

a

)

) and multiplication the

functional

composition

.
It is the ring of

operators

over

A

.

By contrast, the set of all functions

{

f

:

A

→

A

}

are closed to addition and composition, however,
there are generally functions

f

such that

f

∘

(

g

+

h

)

≠

f

∘

g

+

f

∘

g

and so this set
forms only a near ring.

Change of rings (rings generated from other rings)

Let

R

be a ring.

1.

If

I

is an ideal of

R

, then the

quotient

R

/

I

is a ring, called a

quotient ring

.

2.

R

⁢

[

x

]

is the

polynomial ring

over

R

in one indeterminate

x

(or alternatively, one can think that

R

⁢

[

x

]

is any

transcendental extension

ring of

R

, such as

ℤ

⁢

[

π

]

is over

ℤ

),

3.

R

⁢

(

x

)

is the

field of rational functions

in

x

,

4.

R

⁢

[

[

x

]

]

is the ring of

formal power series

in

x

,

5.

R

⁢

(

(

x

)

)

is the ring of formal Laurent series in

x

,

6.

M

n

×

n

⁢

(

R

)

is the

n

×

n

matrix ring

over

R

.

7.

A special case of Example 6 under the

section

on non-commutative rings is the

ring of endomorphisms

over a ring

R

.

8.

For any group

G

, the

group ring

R

⁢

[

G

]

is the set of formal sums of elements of

G

with

coefficients

in

R

.

9.

For any non-empty set

M

and a ring

R

, the set

R

M

of all functions from

M

to

R

may be made a ring

(

R

M

,

+

,

⋅

)

by setting for such functions

f

and

g

(

f

+

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

+

g

⁢

(

x

)

,

(

f

⁢

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

∀

x

∈

M

.

This ring is the often denoted

⊕

M

R

. For

instance

, if

M

=

{

1

,

2

}

, then

R

M

≅

R

⊕

R

.

10.

If

R

is commutative, the

ring of fractions

S

-

1

⁢

R

where

S

is a

multiplicative subset

of

R

not containing 0.

11.

Let

S

,

T

be

subrings

of

R

. Then

(

S

R

0

T

)

:=

{

(

s

r

0

t

)

∣

r

∈

R

,

s

∈

S

,

t

∈

T

}

with the usual

matrix addition

and multiplication is a ring.

Title

examples of rings

Canonical name

ExamplesOfRings

Date of creation

2013-03-22 15:00:42

Last modified on

2013-03-22 15:00:42

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

42

Author

matte (1858)

Entry type

Example

Classification

msc 16-00

Classification

msc 13-00

Related topic

CommutativeRing

Related topic

Ring

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