Advice for computing the Jacobson radical of rings Could you please assist me the way how to compute radical of the rings given below. Is there some simple techniques how I could use easily?
a) $ \mathbb{Z} / n \mathbb{Z} $ of modulo $n$ congruence class
b) Polynomial ring $F[X]$ over field $F$
c)The ring of $n \times n $ upper triangular matrix over a field $F$
 A: Well, some general tips:


*

*If your ring is commutative and finite, then the Jacobson radical is just going to be the set of nilpotent elements.

*If your ring is just commutative, then it will contain the nilpotent elements, but may possibly be bigger.

*The Jacobson radical contains every nilpotent ideal. (meaning $I^n=\{0\}$.

*If $R/I$ is a ring that you know has radical zero, then $J(R)\subseteq I$.
So my hints on your questions are these:


*

*This depends directly on $n$. The first hint above is the easiest thing to apply.

*There is a theorem that tells you the Jacobson radical of $F[X]$ equals the nilradical, as well. I'm not sure there's an easy way to compute nilpotent polynomials in general, but of course when $F$ is a field $F[x]$ is a domain so...

*The strictly upper triangular matrices are a nilpotent ideal, so you know it is contained in $J(R)$. And the quotient by this ideal is a product of fields, which has Jacobson radical zero. So the Jacobson radical of the triangular matrices is the ideal of strictly upper triangular matrices.
