Prime and jacoboson radical

In summary, the conversation revolves around finding the prime and Jacobson radical ideals in the ring Z[T]/(T^3), where Z represents the set of integers. It is discussed that the ring is not a field due to the irreducibility of T^3 over Z[T]. It is also mentioned that nilpotent elements play a key role in determining prime ideals. The conversation ends with a suggestion to look at nilpotent elements when searching for prime ideals.
  • #1
peteryellow
47
0
I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.

Is it true that Z[T]/(T^3) is a field, because T^3 is irreducibel over Z[T]. If it is true that
Z[T]/(T^3) is a field then 0 is the prime and jacobson radical radical.

Is it true please help.
 
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  • #2
No T^3 is not irreducible, so can some body help me that how does prime and maximal ideals in Z[T]/(T^3) look like.
 
  • #3
Think about this abstractly: if R is a ring with an ideal I, what can you say about the ideal structure of R/I?
 
  • #4
if x is nilpotent, i.e. some positive power of x is zero, then what prime ideals does x lie in?what about the converse question? if x is not nilpotent, can you find a prime ideal not containing x? (do you know about localizing a ring at powers of an element?)
 
  • #5
No I don't understand what you are saying mathwonk. My definition of jacobson radical is that it is intersection of maximal ideals and prime radical iks intersection of prime radical.
 
  • #6
mathwonk is alluding to the fact that the prime radical of (a commutative ring) R is nothing but the set of all nilpotent elements in R. The same comment applies to the Jacobson radical of R whenever R is finitely-generated (as a Z-module).
 
  • #7
But Morphism can you please tell me that what are prime and maximal ideals of the ring, and how can I FIND THEM. PLEASE HELP. THNAKS
 
  • #8
But why? mathwonk's method is much more elegant.
 
  • #9
the point is that if you are looking for prime ideals you should look at nilpotent elements.
 

1. What is the definition of prime radical and jacobson radical?

The prime radical of a ring is the intersection of all prime ideals of the ring. The Jacobson radical of a ring is the intersection of all maximal ideals of the ring.

2. What is the importance of prime and jacobson radicals?

Prime and Jacobson radicals are important concepts in ring theory as they provide information about the structure and properties of a ring. They are also used in the classification of rings and in proving theorems about rings.

3. How are prime and jacobson radicals related?

The prime radical of a ring is contained in the Jacobson radical, meaning that every prime ideal is also a maximal ideal. In some cases, the prime and Jacobson radicals may be equal, but this is not always the case.

4. Can prime and jacobson radicals be zero?

Yes, it is possible for the prime and Jacobson radicals to be equal to zero. This happens when the ring has no prime or maximal ideals, such as in the ring of integers.

5. How can prime and jacobson radicals be computed?

There is no general method for computing prime and Jacobson radicals, but there are specific methods for certain types of rings. In some cases, the prime and Jacobson radicals can be determined by analyzing the ring's structure or using other techniques, such as the Artin-Wedderburn theorem.

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