I have the following exercise: Find the radical of the subring $S\subset M_n(\mathbb{C})$, where $S$ consists of upper triangular matrices.
Unfortunately it is not specified what kind of radical is ment. What makes the most sense here? The Jacobson radical?
The Jacobson radical, most likely. It is the set of strictly upper triangular matrices.
Work to confirm this using whatever characterizations of the Jacobson radical that you know.