Nilradical and Ideal Relationship in Commutative Rings: A Mathematical Analysis

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Homework Help Overview

The discussion revolves around the relationship between the ideal \(\sqrt{N}\) in a commutative ring \(R\) and the nilradical of the quotient ring \(R/N\). Participants are exploring definitions and clarifications regarding these mathematical concepts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the definitions of the nilradical and the radical of an ideal. There is a focus on whether the two concepts can be considered equivalent and the implications of their definitions in the context of commutative rings.

Discussion Status

The discussion is ongoing, with participants questioning the accuracy of definitions and the wording of their answers. Some have offered clarifications regarding the terms used, while others are seeking confirmation on their interpretations and the correctness of their statements.

Contextual Notes

There is a noted confusion regarding the definitions of the nilradical and the radical of an ideal, with participants reflecting on the potential discrepancies between their understanding and what was taught in class.

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Homework Statement


Let N be an ideal in of a commutative ring R. What is the relationship of the ideal \sqrt{N} 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?
 
Last edited:
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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 \sqrt{N} 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.
 
Last edited:
Sorry. I was totally wrong.

The radical of an ideal N is defined as the set \sqrt{N} 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...
 
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...
 
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"?
 
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?
 

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