Nilradical and Ideal Relationship in Commutative Rings: A Mathematical Analysis

[ehrenfest]

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?

 

[Hurkyl]

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.

 

[ehrenfest]

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...

 

[ehrenfest]

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...

 

[morphism]

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"?

 

[ehrenfest]

"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?