# Radical of an Ideal

In this installation, we will study more on ideals of a ring *A*.

Definition.If is an ideal, its

radicalis defined by

To fix ideas, again consider the case again. For the ideal (*m*) where , each , its radical is simply (*m’*) where with the repeated exponents removed. Thus one thinks of the radical as “retaining only the prime factors”.

Our first result is:

Lemma.The radical of an ideal is also an ideal.

**Proof**

Let . Suppose ; pick *m*, *n* > 0 such that .

As before, is a sum of terms with and so each term is a multiple of or . Thus so .

Also for any we have . Hence . Finally since , we see that is an ideal of *A*. ♦

# Properties of Radical

The following properties relate the radical of an ideal to earlier constructions.

Proposition.Let be ideals of A, and be any collection of ideals of A.

- .
- .
- .

**Proof**

Since we have proven ⊇ of the first claim. Conversely, if then for some *n* > 0 and so for some *m*, *n* > 0. Thus .

For second claim, since , ⊆ is obvious. Conversely, if *x* lies in the RHS then for some *n* > 0, and so with . Without loss of generality, there is an *m* > 0 such that for each *j* = 1, …, *k* (take *m* large enough). Then

= sum of terms of the form with

In each term, we have for some *j*, hence the term is a multiple of . Thus and *x* lies in the LHS.

Finally for the last claim.

- Since the second term is contained in the first.
- Since we have and similarly so the first term is contained in the third.
- Finally if there exist
*m*,*n*> 0 such that . Then so the third term is contained in the second. ♦

It is not true that for any class of ideals . For example, take and for . Then so

Definition.An ideal is called a

radical idealif .

Note that for any ideal , is a radical ideal.

### Exercise.

1. Prove that a prime ideal is radical.

2. Decide which of the following is true. Find counter-examples for the false claims.

- If are radical ideals, so is .
- If are radical ideals, so is .
- If are radical ideals, so is .
- If is a collection of radical ideals, so is .
- If is a collection of radical ideals, so is .

### Hint (highlight to read)

[For a counter-example to the first claim, take the ring *A* = ℤ[*X*], the ring of polynomials with integer coefficients.]

# Division of Ideals

Finally, we wish to divide ideal by .

Definition.Let be ideals. Write

Here, the notation means ; note that this is an ideal of *A*. As a convenient mnemonic for the definition (whether it is or ), just recall that in the ring ℤ we have (*mn**ℤ : n**ℤ) = m*ℤ.

Lemma.The set is an ideal of A.

**Proof**

Clearly since .

Next suppose , so . Then .

Finally if , so , then any gives us since is an ideal of *A*. ♦

Finally, we go through some basic properties of ideal division.

PropositionLet be ideals of A, and be any collection of ideals of A.

- .
- .
- .
- .

**Proof**

First claim: if , then by definition . Hence also contains any finite sum of with , .

Second claim: for ,

Third claim: for ,

Fourth claim: where the second equivalence follows from: and for any collection of ideals , . ♦

### Note

In the next article, we will be looking at some basic ideas in algebraic geometry to motivate many of our subsequent concepts. The concept of a radical ideal is of paramount importance there. We will not be seeing much of ideal division for a while, until we encounter invertible ideals.