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
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 $\displaystyle R=\frac{\mathbb{Z}[T]}{<T^3>}.$ recall that $\displaystyle N(R)$, prime radical, is also the set of all nilpotent elements of R. now it should be obvious to you that $\displaystyle N(R)=\frac{<T>}{<T^3>}.$
to find $\displaystyle J(R)$, Jacobson radical, we use this fact that in a (commutative) ring S, $\displaystyle a \in J(S),$ if and only if $\displaystyle 1-ax$ is a unit for all $\displaystyle x \in S.$ it's very easy to
see that the units of $\displaystyle R$ are $\displaystyle \pm 1 + \alpha T + \beta T^2 + <T^3>, \ \ \alpha, \beta \in \mathbb{Z}.$ from here conclude that $\displaystyle J(R)=\frac{<T>}{<T^3>}. \ \ \ \square$
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.
an element of $\displaystyle R$ is in the form $\displaystyle u=p(T) + <T^3>,$ where $\displaystyle p(T) \in \mathbb{Z}[T].$ ( by the way we may assume that $\displaystyle p(T)$ is of degree at most $\displaystyle 2$ because
the terms of degree 3 or bigger belong to $\displaystyle <T^3>. $ ) now $\displaystyle u^n = 0$ means $\displaystyle (p(T))^n \in <T^3>,$ which is possible for some $\displaystyle n$ if and only if the constant
term of $\displaystyle p(T)$ is $\displaystyle 0.$ thus $\displaystyle u$ is nilpotent if and only if $\displaystyle p(T) \in <T>.$
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.