Radical of An Ideal - Examples

Examples

Consider the ring Z of integers.

  1. The radical of the ideal 4Z of integer multiples of 4 is 2Z.
  2. The radical of 5Z is 5Z.
  3. The radical of 12Z is 6Z.
  4. In general, the radical of mZ is rZ, where r is the product of all prime factors of m (see radical of an integer). In fact, this generalizes to an arbitrary ideal; see the properties section.

The radical of a primary ideal is prime. If the radical of an ideal I is maximal, then I is primary.

A prime ideal is a radical ideal. In fact, for any prime ideal P.

Let I, J be ideals of a ring R. If are comaximal, then are comaximal.

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