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principal ideals
#The ideal is closed under addition
#The ideal is closed under addition
#The product of an element of the ideal and an element of the initial ring is an element of the ideal
#The product of an element of the ideal and an element of the initial ring is an element of the ideal
In general, one must distinguish between left and right ideals, because many rings do not have commutative multiplication. All discussion here applies to commutative rings.
== Examples ==
== Examples ==
# The sum of two even integers is even.
# The sum of two even integers is even.
# The product of an even integer with and any other integer is even.
# The product of an even integer with and any other integer is even.
This example is itself an example of a '''principal ideal'''. Given any ring R and element <math>x \in R</math>, we may define an ideal (r), which consists of all elements of R which may be written as the product of r with some other element a of the ring. The ideal (r) is called a principal ideal. It is an ideal, because:
# <math> 0 = 0 \cdot r \in (r)</math>
# If <math>x \in (r)</math> and <math>y \in (r)</math>, then we can write <math>x = ra</math> and <math>y = rb</math>. Then <math>x+y = ra + rb = r(a+b) \in R</math>.
# If <math>x \in (r)</math> and <math>b \in R</math> is an arbitrary element of the ring <math>R</math>, then <math>xb = (ar)b = (ab)r \in R</math>.
[[Category:Mathematics]]
[[Category:Mathematics]]