Is Z[T]/(T^3) a Field Due to T^3's Irreducibility?

[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.

Is it true please help.

 

[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]

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]

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 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.

 

[morphism]

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]

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

 

[mathwonk]

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