radical (idea)

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 a^rn^m-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=a^mr^m will lie in I for large enough m.

Examples In the ring of integers Z the ideal 20Z has radical 10Z. In general

rad(p₁^n₁...p_t^n_tZ) = p₁...p_tZ

for distinct primes p_i and positive integers n_i.