Lower triangular matrix

A lower triangular matrix has zero entries above its main diagonal. Its determinant is the product of its diagonal entries, and an invertible lower triangular linear system is solved by forward substitution.

The algebra $T_n(k)$ consists of the lower triangular $n$ by $n$ matrices over $k$. Its Jacobson radical $N$ consists of the strictly lower triangular matrices, andThe $n$ simple left modules $S_r$ are one-dimensional, with a matrix acting through its $r$th diagonal entry. For the left regular module,