# Primary Ideals, prime ideals and maximal ideals - D&F Section 15.2

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In summary: Proposition 19 Part (5) from Dummit and Foote's Section 15.2. The proposition states that if M is a maximal ideal and Q is an ideal with M^n \subseteq Q \subseteq M for some n \geq 1, then Q is a primary ideal with rad Q = M. The proof of this proposition involves showing that rad M = M, which is possible because M is a maximal ideal and thus cannot be contained in any other ideal. This also means that rad M cannot be equal to R, as that would imply M = R, which is not possible for a maximal ideal.
Math Amateur
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MHB
I am studying Dummit and Foote Section 15.2. I am trying to understand the proof of Proposition 19 Part (5) on page 682 (see attachment)

Proposition 19 Part (5) reads as follows:
----------------------------------------------------------------------------------------------------------------------------

Proposition 19.

... ...

(5) Suppose M is a maximal ideal and Q is an ideal with [TEX] M^n \subseteq Q \subseteq M [/TEX] for some [TEX] n \ge 1 [/TEX].

Then Q is a primary idea, with rad Q = M

------------------------------------------------------------------------------------------------------------------------------The proof of (5) above reads as follows:-------------------------------------------------------------------------------------------------------------------------------

Proof.

Suppose [TEX] M^n \subseteq Q \subseteq M [/TEX] for some [TEX] n \ge 1 [/TEX] where M is a maximal idea.

Then [TEX] Q \subseteq M [/TEX] so [TEX] rad \ Q \subseteq rad \ M = M [/TEX].

... ... etc

--------------------------------------------------------------------------------------------------------------------------------

My problem is as follows:

Why can we be sure that rad M = M?

I know that M is maximal and so no ideal in R can contain M. We also know that [TEX] M \subseteq rad \ M [/TEX]

Thus either rad M = M (the conclusion D&F use) or rad M = R?

How do we know that [TEX] rad \ M \ne R [/TEX]?

Would appreciate some help.

Peter

Last edited:
Like you said, we have $$M \subseteq \mbox{Rad} \ M$$ which implies that $$M = \mbox{Rad}\ M$$ as $$M$$ is a maximal ideal. Why is $$\mbox{Rad} \ M \neq R$$? Suppose that $$\mbox{Rad} \ M = R$$ then $$M=R$$ but that's impossible by definition of a maximal ideal.

Siron said:
Like you said, we have $$M \subseteq \mbox{Rad} \ M$$ which implies that $$M = \mbox{Rad}\ M$$ as $$M$$ is a maximal ideal. Why is $$\mbox{Rad} \ M \neq R$$? Suppose that $$\mbox{Rad} \ M = R$$ then $$M=R$$ but that's impossible by definition of a maximal ideal.

Thanks for the helpful post, Siron

Peter

## 1. What is the difference between a primary ideal, prime ideal, and maximal ideal?

A primary ideal is a proper ideal in a ring where any zero-divisor lies in its radical. A prime ideal is a proper ideal in a ring where the product of any two elements in the ring lies in the ideal. A maximal ideal is an ideal that is not properly contained in any other proper ideal.

## 2. How do primary ideals, prime ideals, and maximal ideals relate to each other?

Every maximal ideal is a prime ideal, and every prime ideal is a primary ideal. However, the converse is not always true. In general, a primary ideal is contained in a prime ideal, which is contained in a maximal ideal.

## 3. What is the significance of primary ideals, prime ideals, and maximal ideals in ring theory?

These types of ideals help us understand the structure of a ring and its elements. They also have important applications in algebraic geometry, commutative algebra, and algebraic number theory.

## 4. Can a primary ideal be maximal?

No, a primary ideal cannot be maximal. This is because if an ideal is maximal, it means it is not properly contained in any other proper ideal. However, a primary ideal is always contained in a prime ideal, which is itself contained in a maximal ideal.

## 5. How are primary ideals, prime ideals, and maximal ideals related to zero-divisors in a ring?

A primary ideal is closely related to zero-divisors, as it is an ideal where all zero-divisors in the ring lie in its radical. Prime ideals and maximal ideals are also related to zero-divisors, as they are ideals that help us understand and classify zero-divisors in a ring.

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