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 re ...
, 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 considere ...
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 sp ...
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 bicondi ...
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 In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
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 with ...
.
This concept is generalized to
non-commutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...
s in the
Semiprime ring
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced ...
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 sp ...
, denoted by
or
, is defined as
:
(note that
).
Intuitively,
is obtained by taking all roots of elements of
within the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
. 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 class ...
elements (the
nilradical) of the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
(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
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
(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-Noether ...
, 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
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced r ...
''.
Examples
* Consider the ring
of
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
.
*# 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 {{no footnotes, date=December 2015
In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''.
A ...
factor of
(see
Radical of an integer In number theory, the radical of a positive integer ''n'' is defined as the product of the distinct prime numbers dividing ''n''. Each prime factor of ''n'' occurs exactly once as a factor of this product:
\displaystyle\mathrm(n)=\prod_p
The radic ...
). 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 and ...
section).
* Consider the ideal