The point of this post is to provide examples and non-examples for the following ring classes:
- Ring
- Commutative Ring
- Integral Domain
- Integrally Closed Domain
- Unique Factorization Domain (UFD)
- Principal Ideal Domain (PID)
- Euclidean Domain
- Field
This order forms an inclusion that goes from up to down. That is, the set of rings contains the set of commutative rings, the set of UFDs contains the set of PIDs, etc.
First we define a ring to be a non-empty set, together with two operations called addition and multiplication, denoted by
and
respectively, which satisfy the following axioms:
is an abelian group with respect to
with identity element 0, i.e.:
- Closure under addition.
- Associativity under addition.
- Commutative under addition.
- Existence of additive identity 0.
- Existence of additive inverse
such that
for all
.
- Multiplication
is closed, associative, and has identity element 1 satisfying
for all
. (However, it does not require inverse or commutativity. Beware that some definitions do not require multiplicative identity.)
- The left and right distributive laws hold, i.e.
.
.
Many familiar spaces are rings. In fact, by the inclusion chain, every object given below is a ring.
1. Rings Commutative Rings
A commutative ring is a ring where multiplication is commutative.
Example of a ring which is not a commutative ring:
- Matrix multiplication is not commutative. For example,
, while
.
2. Commutative Rings Integral Domains
An integral domain, also called a commutative domain, is a commutative ring that is also a domain. That is, it contains no zero-divisors. So if and
, then
. Note that our definition is consistent as we require the existence of multiplicative identity in a ring.
Examples of commutative rings which are not integral domains:
- The quotient rings
where
is a composite number. Let
, where
. Then
.
- The continuous functions on
. Let
be
on
and
on
, and let
be
on
and
on
. Then
on
even though neither
nor
is the zero function.
3. Integral Domains Integrally Closed Domains
An integrally closed domain is an integral domain that is integrally closed in its field of fractions.
Examples of integral domains which are not integrally closed domains:
. For instance, the element
is integral over
and is in the quotient field, but is not in
. This is due to
being a finite index subgroup of
. In particular, it has index 2. (On the other hand,
is integrally closed in its quotient field. Note that
is a sixth root of unity and is thus algebraic, whereas
is not algebraic.)
, as it is a subring of the polynomial ring
generated by
and
. There is a singularity at
.
4. Integrally Closed Domains Unique Factorization Domains
A unique factorization domain (UFD) is a commutative domain that satisfies:
- Every non-zero, non-unit of
is a product of a finite number of irreducible elements.
- If
is a non-zero, non-unit of
with
for some units
and irreducibles
, then
,
- There exists a one-to-one and onto function
and units
such that
for
.
Example of an integrally closed domain which is not a UFD:
. For instance, the number
can be factored the usual way as
, or by the unusual way as
. These are different factorizations as the only units in
are
and
.
5. Unique Factorization Domains Principal Ideal Domains
A principal ideal domain (PID) is a commutative domain where every ideal is principal.
Examples of UFDs which are not PIDs:
, the polynomials with coefficients in
.
, the polynomial ring in
variables, if
.
6. Principal Ideal Domains Euclidean Domains
A euclidean domain is a commutative domain such that there exists a function which takes on non-negative values and for every
with
, there exist
such that
with either
or
.
Examples of PIDs which are not euclidean domains:
- The ring of integers of
where
.
7. Euclidean Domains Fields
A field is a commutative ring that has a multiplicative inverse for every non-zero element.
Examples of euclidean domains which are not fields:
- The integers
. The only integers with inverses in
are
and
.
, the formal polynomials over a field
.
, the formal power series over a field
.
Examples of fields:
- The finite fields
, with characteristic
a prime number.
, the set of numbers in the form
, where
and
are rational.
- The rational numbers
.
- The real numbers
.
- The complex numbers
.
By the inclusion chain, fields also belong in every ring class given above.
More Info
For more information about certain types of rings, check out the following:
- Wiki: http://en.wikipedia.org/wiki/Unique_factorization_domain
- Wiki: http://en.wikipedia.org/wiki/Principal_ideal_domain
For more information about the background, see:
Thanks for the post! Currently constructing Z, Q, and R in my intro to advanced math class, and it’s very interesting to see where these concepts are going to fit in next semester in Algebraic Structures when rings are introduced. Just yesterday my professor ended class having proved that R is a linearly ordered field. Happy coincidence!
Keep the posts coming!
Yep, formally constructing fields like
,
, and
for the first time is pretty mind-blowing. It’s like they just naturally arise from the axioms.
Good luck in your class!