In ring theory, a branch of mathematics, the radical of an ideal
of a commutative ring is another ideal defined by the property that an element
is in the radical if and only if some power of
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.
This concept is generalized to non-commutative rings in the semiprime ring article.
is obtained by taking all roots of elements of
is the preimage of the ideal of nilpotent elements (the nilradical) of the quotient ring
(via the natural map
[Note 1] If the radical of
is finitely generated, then some power of
are ideals of a Noetherian ring, then
contains some power of
contains some power of
coincides with its own radical, then
is called a radical ideal or semiprime ideal.
This section will continue the convention that I is an ideal of a commutative ring
: The primary motivation in studying radicals is Hilbert's Nullstellensatz in commutative algebra.
One version of this celebrated theorem states that for any ideal
in the polynomial ring
over an algebraically closed field
, one has where and Geometrically, this says that if a variety
is cut out by the polynomial equations
is a closure operator on the set of ideals of a ring.