Matrix Multiplication over Order n Square Matrices is Closed

Theorem
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathcal M_R \left({n}\right)$ be a $n \times n$ matrix space over $R$.

Then matrix multiplication (conventional) over $\mathcal M_R \left({n}\right)$ is closed.

Proof
From the definition of matrix multiplication, the product of two matrices is another matrix.

The dimensions of an $m \times n$ multiplied by an $n \times p$ matrix is an $m \times p$ matrix, all of whose elements are elements of the ring over which the matrix is formed.

Thus an $n \times n$ multiplied by another $n \times n$ matrix gives another $n \times n$ matrix.

Hence the result.