In
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a branch of
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 ...
, the radical of 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 ...
of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is another ideal defined by the property that an element
is in the radical
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
some power of
is in
. Taking the radical of an ideal is called ''radicalization''. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a
primary ideal is a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
.
This concept is generalized to
non-commutative rings in the
Semiprime ring article.
Definition
The radical of an ideal
in a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, denoted by
or
, is defined as
:
(note that
).
Intuitively,
is obtained by taking all roots of elements of
within the
ring . Equivalently,
is the
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of the ideal of
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cl ...
elements (the
nilradical) of the
quotient ring (via the natural map
). The latter
proves that
is an ideal.
[Here is a direct proof that is an ideal. Start with with some powers . To show that , we use the binomial theorem (which holds for any commutative ring):
:
For each , we have either or . Thus, in each term , one of the exponents will be large enough to make that factor lie in . Since any element of times an element of lies in (as is an ideal), this term lies in . Hence , and so .
To finish checking that the radical is an ideal, take with , and any . Then , so . Thus the radical is an ideal.]
If the radical of
is
finitely generated, then some power of
is contained in
. In particular, if
and
are ideals of a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, then
and
have the same radical if and only if
contains some power of
and
contains some power of
.
If an ideal
coincides with its own radical, then
is called a ''radical ideal'' or ''
semiprime ideal''.
Examples
* Consider the ring
of
integers.
*# The radical of the ideal
of integer multiples of
is
.
*# The radical of
is
.
*# The radical of
is
.
*# In general, the radical of
is
, where
is the product of all distinct
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s of
, the largest
square-free factor of
(see
Radical of an integer). In fact, this generalizes to an arbitrary ideal (see the
Properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy an ...
section).
* Consider the ideal