Graphics Reference
In-Depth Information
(
)
ab
∫
mod
I
,
if a - b ΠI.
It is easy to show that ∫ is an equivalence relation on R and the equivalence classes
are just the cosets of the additive subgroup of R.
Definition.
Let r
1
, r
2
,..., and r
k
be elements of a commutative ring R. then
{
}
<
rr
,
,...,
r
> =
ar ar
+
+◊◊◊+
ar a
Œ
R
12
k
11
22
k k
i
is called the
ideal generated by the r
i
.
It is easy to show that <r
1
,r
2
,...,r
k
> is an ideal.
Definition.
If r is an element of a commutative ring R, then <r> is called the
prin-
cipal ideal generated by r
in R.
Definition.
An ideal I in a ring R, I π R, is called a
maximal
ideal if, whenever J is
an ideal of R with I Õ J Õ R, then either J = I or J = R.
Definition.
A commutative ring R, I π R, with identity is said to be an
integral domain
if ab = 0 implies that either a = 0 or b = 0.
Definition.
An integral domain is called a
principal ideal domain
(PID) if all of its
ideals are principal ideals.
The ring of integers is a good example of a principal ideal domain.
Definition.
Let a and b be elements of a commutative ring R. We say that b
divides
a, denoted by b|a, if b is nonzero and a = bc, for some c in R. One calls b a
factor
or
divisor
of a in this case.
Definition.
Let R be a commutative ring with unity. Two elements a and b in R are
said to be
associates
if a = ub, where u is a unit.
Definition.
Let R be an integral domain. An element a in R is said to be
irreducible
if
(1) a is not 0,
(2) a is not a unit,
(3) for all b in R, if b|a, then b is a unit or b is an associate of a.
Definition.
An integral domain is called a
unique factorization domain
(UFD)
provided that
(1) Every nonzero element r is either a unit or a product of irreducible
elements.
(2) If
rpp
=
◊◊◊
p qq
=
◊◊◊
q
,
12
n
12
m
