Prime and jacoboson radical

peteryellow

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.

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peteryellow

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.

morphism

Homework Helper
Think about this abstractly: if R is a ring with an ideal I, what can you say about the ideal structure of R/I?

mathwonk

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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?)

peteryellow

No I dont 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.

morphism

Homework Helper
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).

peteryellow

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

morphism

Homework Helper
But why? mathwonk's method is much more elegant.

mathwonk

Homework Helper
the point is that if you are looking for prime ideals you should look at nilpotent elements.

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