Let
I be an
ideal in a
commutative ring R. The
radical
of
I is defined to be
rad(I) = { an : a is in I and n is a positive integer }.
Note that
I lies inside its radical (take
n=1). An ideal
that equals its radical is called a
radical ideal. A commutative
ring is called
reduced if the zero ideal is radical.
Lemma
rad(I) is itself an ideal of R.
Proof: For if
a,b are in rad(I) and r is in R
then think about (a-b)m. Expand this using the
binomial theorem and we get a linear combination of terms like
arnm-r. As long as m is big enough
either the power of a or the power of b will lie
I. Hence, so does (a-b)m.
Likewise (ar)m=amrm will lie
in I for large enough m.
Examples
In the ring of integers Z the ideal 20Z has radical
10Z. In general
rad(p1n1...ptntZ) =
p1...ptZ
for distinct
primes
pi and positive integers
ni.