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 $F$

• what is radical? – Raghukul Raman Oct 5 '17 at 13:48
• The Jacobson radical, probably? – rschwieb Oct 5 '17 at 13:49

Well, some general tips:

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

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

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

4. 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:

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

2. 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...

3. 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.