For $1\leq r\leq n$, let $S_r=k$ with a lower triangular matrix $a=(a_{ij})$ acting by the scalar $a_{rr}$. These are pairwise nonisomorphic <simple modules>, since the diagonal <matrix units> $e_{ii}$ distinguish them.

Let $N$ be the ideal of strictly lower triangular matrices. It is nilpotent, and
$$
A/N\cong k^n
$$
is a <semisimple algebra>. Hence $N$ is the <Jacobson radical>; explicitly,
$$
\boxed{J(A)=N=\operatorname{span}_k\{e_{rs}:r>s\}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 1 / d / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-138/1/d/i/solution