prime and jacobson radical, please help

peteryellow · #1 $Old$ 08-28-2008, 06:21 AM

I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.

Please help! Thanks

NonCommAlg · #2 $Old$ 08-28-2008, 04:39 PM

Quote:

Originally Posted by peteryellow $View Post$

I want to find prime and jacobson radical radicals in Z[T]/(T^3), here Z = integers.
My definition of jacobson radical is that it is intersection of maximal ideals and prime radical is intersection of prime ideals.
Please help! Thanks

let $R=\frac{\mathbb{Z}[T]}{<T^3>}.$ recall that $N(R)$ , prime radical, is also the set of all nilpotent elements of R. now it should be obvious to you that $N(R)=\frac{<T>}{<T^3>}.$

to find $J(R)$ , Jacobson radical, we use this fact that in a (commutative) ring S, $a \in J(S),$ if and only if $1-ax$ is a unit for all $x \in S.$ it's very easy to

see that the units of $R$ are $\pm 1 + \alpha T + \beta T^2 + <T^3>, \ \ \alpha, \beta \in \mathbb{Z}.$ from here conclude that $J(R)=\frac{<T>}{<T^3>}. \ \ \ \square$

peteryellow · #3 $Old$ 08-29-2008, 09:15 AM

you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.

and how do you know that $\pm 1 + \alphaT + \betaT + <T^3>$ is a unit in R but if this is a unit how can you conclude that jacobson radical is $<T>/<T^3>$.
please help thnaks.

NonCommAlg · #4 $Old$ 08-29-2008, 02:03 PM

Quote:

Originally Posted by peteryellow $View Post$

you are saying that the set of prime radical is same as set of nilpotents, fine but How do you get that the prime radical is $<T>/<T^3>$.

an element of $R$ is in the form $u=p(T) + <T^3>,$ where $p(T) \in \mathbb{Z}[T].$ ( by the way we may assume that $p(T)$ is of degree at most $2$ because

the terms of degree 3 or bigger belong to $<T^3>.$ ) now $u^n = 0$ means $(p(T))^n \in <T^3>,$ which is possible for some $n$ if and only if the constant

term of $p(T)$ is $0.$ thus $u$ is nilpotent if and only if $p(T) \in <T>.$

the second part of your question (Jacobson radical) is as easy as this part. just follow the line of the proof that i gave you.