In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
ring theory, a principal ideal is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
in a
ring that is generated by a single element
of
through multiplication by every element of
The term also has another, similar meaning in
order theory, where it refers to an
(order) ideal in a
poset generated by a single element
which is to say the set of all elements less than or equal to
in
The remainder of this article addresses the ring-theoretic concept.
Definitions
* a ''left principal ideal'' of
is a
subset of
given by
for some element
* a ''right principal ideal'' of
is a subset of
given by
for some element
* a ''two-sided principal ideal'' of
is a subset of
given by
for some element
namely, the set of all finite sums of elements of the form
While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition.
If
is a
commutative ring with identity, then the above three notions are all the same.
In that case, it is common to write the ideal generated by
as
or
Examples of non-principal ideal
Not all ideals are principal.
For example, consider the commutative ring