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 recall that , prime radical, is also the set of all nilpotent elements of R. now it should be obvious to you that
to find , Jacobson radical, we use this fact that in a (commutative) ring S, if and only if is a unit for all it's very easy to
see that the units of are from here conclude that
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 is in the form where ( by the way we may assume that is of degree at most because
the terms of degree 3 or bigger belong to ) now means which is possible for some if and only if the constant
term of is thus is nilpotent if and only if
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.