Nilradical and Ideal Relationship in Commutative Rings: A Mathematical Analysis

  • Thread starter ehrenfest
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In summary, the radical of an ideal N is defined as the set of all elements a in R such that a^n is in N for some positive integer n, while the nilradical of a ring is the collection of all nilpotent elements. Therefore, a is in the radical of N if and only if (a+N) is in the nilradical of R/N. This relationship is often referred to as "canonically homomorphic", although there may be disagreement on whether they can be considered exactly the same. The instruction to word the answer carefully may have been given due to confusion between the definitions of "radical" and "nilradical".
  • #1
ehrenfest
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Homework Statement


Let N be an ideal in of a commutative ring R. What is the relationship of the ideal [itex]\sqrt{N} [/itex] to the nilradical of R/N? Word your answer carefully.

Recall that the nilradical of an ideal N is the collection of all elements a in R such that a^n is in N for some n in Z^+.

EDIT: this definition is dead wrong

Homework Equations


The Attempt at a Solution


Answer: a is in the nilradical of N iff (a+N) is in the nilradical of R/N. So they are the same. Why did they say word your answer carefully?
 
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  • #2
ehrenfest said:
Recall that the nilradical of an ideal N is the collection of all elements a in R such that a^n is in N for some n in Z^+.
Are you quite sure that's the definition of "nilradical" given in your class? The definition you give is indeed the definition of [itex]\sqrt{N}[/itex] but that's called the "radical of N". The term "nilradical" applies to rings, and the nilradical of R is the radical of its zero ideal. (i.e. the set of nilpotent elements of R)

So they are the same.
They're obviously not "the same"; one is an ideal of R, the other is an ideal of R/N, and they (usually) don't have a single element in common.
 
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  • #3
Sorry. I was totally wrong.

The radical of an ideal N is defined as the set [itex] \sqrt{N} [/itex] of all a in R such that a^n is in N for some positive integer n.

The nilradical of a ring is the collection of all the nilpotent elements.

Let me see if I can figure it out with the correct definitions...
 
  • #4
Is this answer worded correctly:

a is in the radical of N iff (a+N) is in the nilradical of R/N

So they are canonically homomorphic, right? I would call them the same but it seems like other people disagree...
 
  • #5
ehrenfest said:
a is in the radical of N iff (a+N) is in the nilradical of R/N
Looks good.

So they are canonically homomorphic, right? I would call them the same but it seems like other people disagree...
"Canonically homomorphic"?
 
  • #6
morphism said:
"Canonically homomorphic"?

OK. Forget that.

My final answer is "a is in the radical of N iff (a+N) is in the nilradical of R/N".

Why did they say word your answer carefully? Is my answer worded carefully enough?
 

FAQ: Nilradical and Ideal Relationship in Commutative Rings: A Mathematical Analysis

What is Nilradical math?

Nilradical math is a concept in abstract algebra that refers to the set of all nilpotent elements of a ring. In simpler terms, it is the collection of all elements in a ring that, when raised to a certain power, become zero.

Why is Nilradical math important?

Nilradical math is important because it helps us understand the structure of rings and their properties. It also plays a crucial role in the study of algebraic geometry and representation theory.

What are some examples of Nilradical math?

Examples of Nilradical math include the zero element in any ring, and in the ring of integers, all multiples of a prime number (such as 2, 3, 5, etc.) are nilpotent elements.

How is Nilradical math related to nilpotent elements?

Nilradical math and nilpotent elements are closely related, as the nilradical of a ring is the set of all nilpotent elements in that ring. In other words, the nilradical is the "smallest" ideal of a ring that contains all nilpotent elements.

What are some properties of Nilradical math?

Some properties of Nilradical math include: it is always an ideal of a ring, it is always contained in the Jacobson radical of a ring, and it is always a prime ideal in a commutative ring.

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