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.

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

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 $\pm 1 + \alpha T + \beta T^2 + , \ \ \alpha, \beta \in \mathbb{Z}.$ from here conclude that $J(R)=\frac{}{}. \ \ \ \square$

3. 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 $<T>/<T^3>$.
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 $u=p(T) + ,$ 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 $.$ ) now $u^n = 0$ means $(p(T))^n \in ,$ 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 $p(T) \in .$