Triangular Matrices forms Subring of Square Matrices

Theorem
Let $\mathcal M_K \left({n}\right)$ be the order $n$ square matrix space over a field $K$.

Let $U_K \left({n}\right)$ be the set of upper triangular matrices of order $n$ over $K$.

Then $U_K \left({n}\right)$ forms a subring of $\mathcal M_K \left({n}\right)$ on the operations of matrix entrywise addition and conventional matrix multiplication.

Similarly, let $L_K \left({n}\right)$ be the set of lower triangular matrices of order $n$ over $K$.

Then $L_K \left({n}\right)$ forms a subring of $\mathcal M_K \left({n}\right)$ on the operations of matrix entrywise addition and conventional matrix multiplication.

Proof
From Negative of Triangular Matrix, if $\mathbf B \in U_K \left({n}\right)$ then $-\mathbf B \in U_K \left({n}\right)$.

Then from Sum of Triangular Matrices, if $\mathbf A, -\mathbf B \in U_K \left({n}\right)$ then $\mathbf A + \left({-\mathbf B}\right) \in U_K \left({n}\right)$.

From Product of Triangular Matrices, if $\mathbf A, \mathbf B \in U_K \left({n}\right)$ then $\mathbf A \mathbf B \in U_K \left({n}\right)$.

The result follows from the Subring Test.

The same argument can be applied to matrices in $L_K \left({n}\right)$.