# Nil Radical Proof: Proving A's Ideal is Non-Empty

• JasonJo
In summary, the conversation is discussing the concept of the nil radical of an ideal A in a commutative ring with unity R. The definition of N(A) is given as the set of all elements r in A such that r^n is also in A, where n is a positive integer that depends on r. It is unclear whether the equation provided is a definition or something to be proven, and clarification is needed on the specific task at hand.
JasonJo

## Homework Statement

Let A be any ideal of a commutative ring with unity R. Show that the nil radical of A, N(A)= {r|r^n is in A} where n is a positive integer, and n depends on r.

## The Attempt at a Solution

i don't quite understand the concept of the nil radical of A, what would be an element of this ideal? the unity element is in for every n a positive integer, so it's non empty but I'm still a little confused.

Is the equation you have a definition, or something you're asked to prove? If it's a definition, then what are you being asked to prove? If you're being asked to prove that equation, then what's the definition of N(A)?

## 1. What is the concept of "Nil Radical" in mathematics?

The Nil Radical of a ring is the set of all elements that become zero when raised to a high enough power. In other words, it is the set of all "nilpotent" elements in a ring.

## 2. Why is it important to prove that A's Ideal is non-empty?

Proving that A's Ideal is non-empty is important because it allows us to establish the existence of a non-zero element in A that becomes zero when raised to a high enough power. This can help us understand the structure and properties of the ring A.

## 3. How can we prove that A's Ideal is non-empty?

To prove that A's Ideal is non-empty, we can use the concept of "Nil Radical" and show that the Nil Radical of A is a subset of A's Ideal. This can be done by showing that any nilpotent element of A belongs to A's Ideal.

## 4. What is the significance of proving A's Ideal is non-empty in ring theory?

In ring theory, proving A's Ideal is non-empty is significant because it helps us understand the structure of rings and their ideals. It also allows us to establish important properties of rings, such as their dimension and algebraic properties.

## 5. Are there any applications of proving A's Ideal is non-empty?

Yes, there are many applications of proving A's Ideal is non-empty, especially in algebraic geometry and commutative algebra. It can also be used in the study of modules and vector spaces, as well as in the characterization of algebraic structures.

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