Near Matches
Ignore Exact
Everything
2
integral domain (thing)
See all of integral domain
, there are 2 more in this node.
(
thing
)
by
Noether
Fri Aug 04 2000 at 13:50:19
A
ring
R
is called an
integral domain
if whenever
a,b
in
R
satisfy
ab=0
we have either
a=0
or
b=0
.
Examples
Any
field
is an integral domain. So this includes
Q
,
R
,
C
, and
Z
p
, the
rational number
s, the
real number
s, the
complex number
s, and the integers modulo a prime
p
.
Any
subring
of a field is an integral domain. So this includes the ring of
integer
s
Z
and the
Gaussian integer
s
Z
[i].
The ring
k
[x] of
polynomial
s over another integral domain
k
form another integral domain (proved by looking at leading terms). So this gives lots more examples, like
Z
[x],
C
[x] or
Q
[x,y]
The ring
Z
m
of integers modulo
m
for
m> 3
composite
is not an integral domain.
The ring of
2x2
matrices
with complex number coefficients is not an integral domain.
Odd-Even Theorem
Ring
subring
maximal ideal
characteristic
field
zero divisor
field of fractions
Gaussian integer
Fundamental theorem of arithmetic
commutative ring
principal ideal
prime ideal
highest common factor
unique factorization domain
principal ideal domain
Euclidean ring
ring theory
RNG
Irreducible
Composite
domain
prime